L 2 – Vectors and Scalars Outline • Physical quantities - vectors and scalars • Addition and subtraction of vector • Resultant vector • Change in a vector quantity, calculating relative and resolve a vector into components • Vector representation in a component form in a coordinate system
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L 2 – Vectors and Scalars Outline Physical quantities - vectors and scalars
L 2 – Vectors and Scalars Outline Physical quantities - vectors and scalars Addition and subtraction of vector Resultant vector Change in a vector quantity, calculating relative and resolve a vector into components V ector representation in a component form in a coordinate system. - PowerPoint PPT Presentation
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L 2 ndash Vectors and Scalars
Outlinebull Physical quantities - vectors and scalarsbull Addition and subtraction of vectorbull Resultant vectorbull Change in a vector quantity calculating relative
and resolve a vector into componentsbull Vector representation in a component form in a
coordinate system
Vectors in Physics-Examples
Many physical quantities have both magnitude and direction they are called vectorsbullExamples displacement velocity acceleration force momentum
Other physical quantities have only magnitude they are called scalars bullExamples distance speed mass energy
Displacement and Distance
bull Displacement is the vector connecting a starting point A and some final point B
A
B
bull Distance is the length one would travel from point A to the final point B Therefore distance is a scalar
Geometrical Representation of Vectors
bull Arrows on a plane or spacebull To indicate a vector we use bold letters or an arrow on top of a letter
Properties of Vectors
1 The opposite of a vector a is vector - a
2 It has the same length but opposite direction
3 Two vectors a and b are parallel if one is a positive multiple of the other
a = m b mgt0
Example
if a = 3 b then a is parallel to b
(if a = -2 b then a is anti-parallel to b)
Operations Adding two Vectors
When we add two vectors we get the resultant vector a + b with the parallelogram rule
Operations Adding more vectors
bull We can add more vectors by pairing themappropriately
Operations Vector Subtraction
bull Special case of vector addition Add the negative of the subtracted
vectorbull a ndash b = a + (ndash b)
Components of a Vector
bull A component is a part or shadow along a given direction
bull It is useful to use rectangular componentsndash These are the
projections of the vector along the x- and y-axes
Components of a Vector cont
bull The x-component of a vector is the projection along the x-axis
ax = a cosθ
bull The y-component of a vector is the projection along the y-axis
ay = a sinθ
bull a is the magnitude of vector a a2 = ax
2 + ay2
Example 1
Resolve this vector along the x and y axes to find its components respectively
Example 2
A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector
150 N
100 N
The Unit Vectors i j k
A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction
The Unit Vectors Magnitude
Example 3 Given the two displacements
kjid 010306
kjic 080504
Show that the magnitude of e is approximately 17 units where cde 0102
bull Any vector a can be written as
a = x i + y j + z k
S - South
N - North
E - EastW - West
15deg
15 deg east of north or 75 deg north of east or
bearing of 15 deg
45 deg west of south or45 deg south of west or
bearing of 225 deg
45deg 30deghellip
hellip
30deg
Direction of Vectors
Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j
3
4
Note This vector could also be written in matrix form
The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by
VCP = VCE - VPE
Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE
A pedestrian (P) is travelling relative to the earth (E) with velocity VPE
Relative velocity
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
Vectors in Physics-Examples
Many physical quantities have both magnitude and direction they are called vectorsbullExamples displacement velocity acceleration force momentum
Other physical quantities have only magnitude they are called scalars bullExamples distance speed mass energy
Displacement and Distance
bull Displacement is the vector connecting a starting point A and some final point B
A
B
bull Distance is the length one would travel from point A to the final point B Therefore distance is a scalar
Geometrical Representation of Vectors
bull Arrows on a plane or spacebull To indicate a vector we use bold letters or an arrow on top of a letter
Properties of Vectors
1 The opposite of a vector a is vector - a
2 It has the same length but opposite direction
3 Two vectors a and b are parallel if one is a positive multiple of the other
a = m b mgt0
Example
if a = 3 b then a is parallel to b
(if a = -2 b then a is anti-parallel to b)
Operations Adding two Vectors
When we add two vectors we get the resultant vector a + b with the parallelogram rule
Operations Adding more vectors
bull We can add more vectors by pairing themappropriately
Operations Vector Subtraction
bull Special case of vector addition Add the negative of the subtracted
vectorbull a ndash b = a + (ndash b)
Components of a Vector
bull A component is a part or shadow along a given direction
bull It is useful to use rectangular componentsndash These are the
projections of the vector along the x- and y-axes
Components of a Vector cont
bull The x-component of a vector is the projection along the x-axis
ax = a cosθ
bull The y-component of a vector is the projection along the y-axis
ay = a sinθ
bull a is the magnitude of vector a a2 = ax
2 + ay2
Example 1
Resolve this vector along the x and y axes to find its components respectively
Example 2
A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector
150 N
100 N
The Unit Vectors i j k
A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction
The Unit Vectors Magnitude
Example 3 Given the two displacements
kjid 010306
kjic 080504
Show that the magnitude of e is approximately 17 units where cde 0102
bull Any vector a can be written as
a = x i + y j + z k
S - South
N - North
E - EastW - West
15deg
15 deg east of north or 75 deg north of east or
bearing of 15 deg
45 deg west of south or45 deg south of west or
bearing of 225 deg
45deg 30deghellip
hellip
30deg
Direction of Vectors
Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j
3
4
Note This vector could also be written in matrix form
The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by
VCP = VCE - VPE
Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE
A pedestrian (P) is travelling relative to the earth (E) with velocity VPE
Relative velocity
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
Displacement and Distance
bull Displacement is the vector connecting a starting point A and some final point B
A
B
bull Distance is the length one would travel from point A to the final point B Therefore distance is a scalar
Geometrical Representation of Vectors
bull Arrows on a plane or spacebull To indicate a vector we use bold letters or an arrow on top of a letter
Properties of Vectors
1 The opposite of a vector a is vector - a
2 It has the same length but opposite direction
3 Two vectors a and b are parallel if one is a positive multiple of the other
a = m b mgt0
Example
if a = 3 b then a is parallel to b
(if a = -2 b then a is anti-parallel to b)
Operations Adding two Vectors
When we add two vectors we get the resultant vector a + b with the parallelogram rule
Operations Adding more vectors
bull We can add more vectors by pairing themappropriately
Operations Vector Subtraction
bull Special case of vector addition Add the negative of the subtracted
vectorbull a ndash b = a + (ndash b)
Components of a Vector
bull A component is a part or shadow along a given direction
bull It is useful to use rectangular componentsndash These are the
projections of the vector along the x- and y-axes
Components of a Vector cont
bull The x-component of a vector is the projection along the x-axis
ax = a cosθ
bull The y-component of a vector is the projection along the y-axis
ay = a sinθ
bull a is the magnitude of vector a a2 = ax
2 + ay2
Example 1
Resolve this vector along the x and y axes to find its components respectively
Example 2
A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector
150 N
100 N
The Unit Vectors i j k
A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction
The Unit Vectors Magnitude
Example 3 Given the two displacements
kjid 010306
kjic 080504
Show that the magnitude of e is approximately 17 units where cde 0102
bull Any vector a can be written as
a = x i + y j + z k
S - South
N - North
E - EastW - West
15deg
15 deg east of north or 75 deg north of east or
bearing of 15 deg
45 deg west of south or45 deg south of west or
bearing of 225 deg
45deg 30deghellip
hellip
30deg
Direction of Vectors
Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j
3
4
Note This vector could also be written in matrix form
The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by
VCP = VCE - VPE
Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE
A pedestrian (P) is travelling relative to the earth (E) with velocity VPE
Relative velocity
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
Geometrical Representation of Vectors
bull Arrows on a plane or spacebull To indicate a vector we use bold letters or an arrow on top of a letter
Properties of Vectors
1 The opposite of a vector a is vector - a
2 It has the same length but opposite direction
3 Two vectors a and b are parallel if one is a positive multiple of the other
a = m b mgt0
Example
if a = 3 b then a is parallel to b
(if a = -2 b then a is anti-parallel to b)
Operations Adding two Vectors
When we add two vectors we get the resultant vector a + b with the parallelogram rule
Operations Adding more vectors
bull We can add more vectors by pairing themappropriately
Operations Vector Subtraction
bull Special case of vector addition Add the negative of the subtracted
vectorbull a ndash b = a + (ndash b)
Components of a Vector
bull A component is a part or shadow along a given direction
bull It is useful to use rectangular componentsndash These are the
projections of the vector along the x- and y-axes
Components of a Vector cont
bull The x-component of a vector is the projection along the x-axis
ax = a cosθ
bull The y-component of a vector is the projection along the y-axis
ay = a sinθ
bull a is the magnitude of vector a a2 = ax
2 + ay2
Example 1
Resolve this vector along the x and y axes to find its components respectively
Example 2
A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector
150 N
100 N
The Unit Vectors i j k
A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction
The Unit Vectors Magnitude
Example 3 Given the two displacements
kjid 010306
kjic 080504
Show that the magnitude of e is approximately 17 units where cde 0102
bull Any vector a can be written as
a = x i + y j + z k
S - South
N - North
E - EastW - West
15deg
15 deg east of north or 75 deg north of east or
bearing of 15 deg
45 deg west of south or45 deg south of west or
bearing of 225 deg
45deg 30deghellip
hellip
30deg
Direction of Vectors
Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j
3
4
Note This vector could also be written in matrix form
The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by
VCP = VCE - VPE
Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE
A pedestrian (P) is travelling relative to the earth (E) with velocity VPE
Relative velocity
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
Properties of Vectors
1 The opposite of a vector a is vector - a
2 It has the same length but opposite direction
3 Two vectors a and b are parallel if one is a positive multiple of the other
a = m b mgt0
Example
if a = 3 b then a is parallel to b
(if a = -2 b then a is anti-parallel to b)
Operations Adding two Vectors
When we add two vectors we get the resultant vector a + b with the parallelogram rule
Operations Adding more vectors
bull We can add more vectors by pairing themappropriately
Operations Vector Subtraction
bull Special case of vector addition Add the negative of the subtracted
vectorbull a ndash b = a + (ndash b)
Components of a Vector
bull A component is a part or shadow along a given direction
bull It is useful to use rectangular componentsndash These are the
projections of the vector along the x- and y-axes
Components of a Vector cont
bull The x-component of a vector is the projection along the x-axis
ax = a cosθ
bull The y-component of a vector is the projection along the y-axis
ay = a sinθ
bull a is the magnitude of vector a a2 = ax
2 + ay2
Example 1
Resolve this vector along the x and y axes to find its components respectively
Example 2
A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector
150 N
100 N
The Unit Vectors i j k
A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction
The Unit Vectors Magnitude
Example 3 Given the two displacements
kjid 010306
kjic 080504
Show that the magnitude of e is approximately 17 units where cde 0102
bull Any vector a can be written as
a = x i + y j + z k
S - South
N - North
E - EastW - West
15deg
15 deg east of north or 75 deg north of east or
bearing of 15 deg
45 deg west of south or45 deg south of west or
bearing of 225 deg
45deg 30deghellip
hellip
30deg
Direction of Vectors
Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j
3
4
Note This vector could also be written in matrix form
The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by
VCP = VCE - VPE
Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE
A pedestrian (P) is travelling relative to the earth (E) with velocity VPE
Relative velocity
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
Operations Adding two Vectors
When we add two vectors we get the resultant vector a + b with the parallelogram rule
Operations Adding more vectors
bull We can add more vectors by pairing themappropriately
Operations Vector Subtraction
bull Special case of vector addition Add the negative of the subtracted
vectorbull a ndash b = a + (ndash b)
Components of a Vector
bull A component is a part or shadow along a given direction
bull It is useful to use rectangular componentsndash These are the
projections of the vector along the x- and y-axes
Components of a Vector cont
bull The x-component of a vector is the projection along the x-axis
ax = a cosθ
bull The y-component of a vector is the projection along the y-axis
ay = a sinθ
bull a is the magnitude of vector a a2 = ax
2 + ay2
Example 1
Resolve this vector along the x and y axes to find its components respectively
Example 2
A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector
150 N
100 N
The Unit Vectors i j k
A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction
The Unit Vectors Magnitude
Example 3 Given the two displacements
kjid 010306
kjic 080504
Show that the magnitude of e is approximately 17 units where cde 0102
bull Any vector a can be written as
a = x i + y j + z k
S - South
N - North
E - EastW - West
15deg
15 deg east of north or 75 deg north of east or
bearing of 15 deg
45 deg west of south or45 deg south of west or
bearing of 225 deg
45deg 30deghellip
hellip
30deg
Direction of Vectors
Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j
3
4
Note This vector could also be written in matrix form
The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by
VCP = VCE - VPE
Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE
A pedestrian (P) is travelling relative to the earth (E) with velocity VPE
Relative velocity
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
Operations Adding more vectors
bull We can add more vectors by pairing themappropriately
Operations Vector Subtraction
bull Special case of vector addition Add the negative of the subtracted
vectorbull a ndash b = a + (ndash b)
Components of a Vector
bull A component is a part or shadow along a given direction
bull It is useful to use rectangular componentsndash These are the
projections of the vector along the x- and y-axes
Components of a Vector cont
bull The x-component of a vector is the projection along the x-axis
ax = a cosθ
bull The y-component of a vector is the projection along the y-axis
ay = a sinθ
bull a is the magnitude of vector a a2 = ax
2 + ay2
Example 1
Resolve this vector along the x and y axes to find its components respectively
Example 2
A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector
150 N
100 N
The Unit Vectors i j k
A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction
The Unit Vectors Magnitude
Example 3 Given the two displacements
kjid 010306
kjic 080504
Show that the magnitude of e is approximately 17 units where cde 0102
bull Any vector a can be written as
a = x i + y j + z k
S - South
N - North
E - EastW - West
15deg
15 deg east of north or 75 deg north of east or
bearing of 15 deg
45 deg west of south or45 deg south of west or
bearing of 225 deg
45deg 30deghellip
hellip
30deg
Direction of Vectors
Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j
3
4
Note This vector could also be written in matrix form
The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by
VCP = VCE - VPE
Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE
A pedestrian (P) is travelling relative to the earth (E) with velocity VPE
Relative velocity
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
Operations Vector Subtraction
bull Special case of vector addition Add the negative of the subtracted
vectorbull a ndash b = a + (ndash b)
Components of a Vector
bull A component is a part or shadow along a given direction
bull It is useful to use rectangular componentsndash These are the
projections of the vector along the x- and y-axes
Components of a Vector cont
bull The x-component of a vector is the projection along the x-axis
ax = a cosθ
bull The y-component of a vector is the projection along the y-axis
ay = a sinθ
bull a is the magnitude of vector a a2 = ax
2 + ay2
Example 1
Resolve this vector along the x and y axes to find its components respectively
Example 2
A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector
150 N
100 N
The Unit Vectors i j k
A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction
The Unit Vectors Magnitude
Example 3 Given the two displacements
kjid 010306
kjic 080504
Show that the magnitude of e is approximately 17 units where cde 0102
bull Any vector a can be written as
a = x i + y j + z k
S - South
N - North
E - EastW - West
15deg
15 deg east of north or 75 deg north of east or
bearing of 15 deg
45 deg west of south or45 deg south of west or
bearing of 225 deg
45deg 30deghellip
hellip
30deg
Direction of Vectors
Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j
3
4
Note This vector could also be written in matrix form
The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by
VCP = VCE - VPE
Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE
A pedestrian (P) is travelling relative to the earth (E) with velocity VPE
Relative velocity
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
Components of a Vector
bull A component is a part or shadow along a given direction
bull It is useful to use rectangular componentsndash These are the
projections of the vector along the x- and y-axes
Components of a Vector cont
bull The x-component of a vector is the projection along the x-axis
ax = a cosθ
bull The y-component of a vector is the projection along the y-axis
ay = a sinθ
bull a is the magnitude of vector a a2 = ax
2 + ay2
Example 1
Resolve this vector along the x and y axes to find its components respectively
Example 2
A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector
150 N
100 N
The Unit Vectors i j k
A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction
The Unit Vectors Magnitude
Example 3 Given the two displacements
kjid 010306
kjic 080504
Show that the magnitude of e is approximately 17 units where cde 0102
bull Any vector a can be written as
a = x i + y j + z k
S - South
N - North
E - EastW - West
15deg
15 deg east of north or 75 deg north of east or
bearing of 15 deg
45 deg west of south or45 deg south of west or
bearing of 225 deg
45deg 30deghellip
hellip
30deg
Direction of Vectors
Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j
3
4
Note This vector could also be written in matrix form
The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by
VCP = VCE - VPE
Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE
A pedestrian (P) is travelling relative to the earth (E) with velocity VPE
Relative velocity
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
Components of a Vector cont
bull The x-component of a vector is the projection along the x-axis
ax = a cosθ
bull The y-component of a vector is the projection along the y-axis
ay = a sinθ
bull a is the magnitude of vector a a2 = ax
2 + ay2
Example 1
Resolve this vector along the x and y axes to find its components respectively
Example 2
A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector
150 N
100 N
The Unit Vectors i j k
A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction
The Unit Vectors Magnitude
Example 3 Given the two displacements
kjid 010306
kjic 080504
Show that the magnitude of e is approximately 17 units where cde 0102
bull Any vector a can be written as
a = x i + y j + z k
S - South
N - North
E - EastW - West
15deg
15 deg east of north or 75 deg north of east or
bearing of 15 deg
45 deg west of south or45 deg south of west or
bearing of 225 deg
45deg 30deghellip
hellip
30deg
Direction of Vectors
Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j
3
4
Note This vector could also be written in matrix form
The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by
VCP = VCE - VPE
Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE
A pedestrian (P) is travelling relative to the earth (E) with velocity VPE
Relative velocity
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
Example 1
Resolve this vector along the x and y axes to find its components respectively
Example 2
A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector
150 N
100 N
The Unit Vectors i j k
A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction
The Unit Vectors Magnitude
Example 3 Given the two displacements
kjid 010306
kjic 080504
Show that the magnitude of e is approximately 17 units where cde 0102
bull Any vector a can be written as
a = x i + y j + z k
S - South
N - North
E - EastW - West
15deg
15 deg east of north or 75 deg north of east or
bearing of 15 deg
45 deg west of south or45 deg south of west or
bearing of 225 deg
45deg 30deghellip
hellip
30deg
Direction of Vectors
Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j
3
4
Note This vector could also be written in matrix form
The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by
VCP = VCE - VPE
Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE
A pedestrian (P) is travelling relative to the earth (E) with velocity VPE
Relative velocity
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
Example 2
A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector
150 N
100 N
The Unit Vectors i j k
A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction
The Unit Vectors Magnitude
Example 3 Given the two displacements
kjid 010306
kjic 080504
Show that the magnitude of e is approximately 17 units where cde 0102
bull Any vector a can be written as
a = x i + y j + z k
S - South
N - North
E - EastW - West
15deg
15 deg east of north or 75 deg north of east or
bearing of 15 deg
45 deg west of south or45 deg south of west or
bearing of 225 deg
45deg 30deghellip
hellip
30deg
Direction of Vectors
Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j
3
4
Note This vector could also be written in matrix form
The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by
VCP = VCE - VPE
Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE
A pedestrian (P) is travelling relative to the earth (E) with velocity VPE
Relative velocity
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
The Unit Vectors i j k
A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction
The Unit Vectors Magnitude
Example 3 Given the two displacements
kjid 010306
kjic 080504
Show that the magnitude of e is approximately 17 units where cde 0102
bull Any vector a can be written as
a = x i + y j + z k
S - South
N - North
E - EastW - West
15deg
15 deg east of north or 75 deg north of east or
bearing of 15 deg
45 deg west of south or45 deg south of west or
bearing of 225 deg
45deg 30deghellip
hellip
30deg
Direction of Vectors
Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j
3
4
Note This vector could also be written in matrix form
The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by
VCP = VCE - VPE
Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE
A pedestrian (P) is travelling relative to the earth (E) with velocity VPE
Relative velocity
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
The Unit Vectors Magnitude
Example 3 Given the two displacements
kjid 010306
kjic 080504
Show that the magnitude of e is approximately 17 units where cde 0102
bull Any vector a can be written as
a = x i + y j + z k
S - South
N - North
E - EastW - West
15deg
15 deg east of north or 75 deg north of east or
bearing of 15 deg
45 deg west of south or45 deg south of west or
bearing of 225 deg
45deg 30deghellip
hellip
30deg
Direction of Vectors
Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j
3
4
Note This vector could also be written in matrix form
The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by
VCP = VCE - VPE
Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE
A pedestrian (P) is travelling relative to the earth (E) with velocity VPE
Relative velocity
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
S - South
N - North
E - EastW - West
15deg
15 deg east of north or 75 deg north of east or
bearing of 15 deg
45 deg west of south or45 deg south of west or
bearing of 225 deg
45deg 30deghellip
hellip
30deg
Direction of Vectors
Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j
3
4
Note This vector could also be written in matrix form
The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by
VCP = VCE - VPE
Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE
A pedestrian (P) is travelling relative to the earth (E) with velocity VPE
Relative velocity
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j
3
4
Note This vector could also be written in matrix form
The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by
VCP = VCE - VPE
Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE
A pedestrian (P) is travelling relative to the earth (E) with velocity VPE
Relative velocity
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by
VCP = VCE - VPE
Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE
A pedestrian (P) is travelling relative to the earth (E) with velocity VPE
Relative velocity
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
Example 5
A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
The dot (scalar) product
bull Imagine two vectors a b at an angle θ
bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
CHECK LISTbull READING
Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53
Summarybull Be able to give examples of physical quantities
represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity
calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in
component form in a coordinate systembull Be able to do calculations which demonstrate that you
have understood the above concepts
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg
PowerPoint Presentation
Vectors in Physics-Examples
Slide 3
Geometrical Representation of Vectors
Properties of Vectors
Operations Adding two Vectors
Operations Adding more vectors
Operations Vector Subtraction
Components of a Vector
Components of a Vector cont
Example 1
Example 2
Slide 13
Slide 14
Slide 15
Example 4
Slide 17
Example 5
The dot (scalar) product
CHECK LIST
Numerical Answers for Examples
Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash
Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)
bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)
bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg