STROUD Worked examples and exercises are in the text Programme 6: Vectors VECTORS PROGRAMME 6
Jan 18, 2018
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
VECTORS
PROGRAMME 6
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
(a) A scalar quantity is defined completely by a single number with appropriate units
(b) A vector quantity is defined completely when we know not only its magnitude (with units) but also the direction in which it operates
Physical quantities can be divided into two main groups, scalar quantities and vector quantities.
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Vector representation
A vector quantity can be represented graphically by a line, drawn so that:
(a) The length of the line denotes the magnitude of the quantity(b) The direction of the line (indicated by an arrowhead) denotes the
direction in which the vector quantity acts.
The vector quantity AB is referred to as or a
____AB
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Vector representation
Two equal vectors
Types of vectors
Addition of vectors
The sum of a number of vectors
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Vector representation
Two equal vectors
If two vectors, a and b, are said to be equal, they have the same magnitude and the same direction
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Vector representation
If two vectors, a and b, have the same magnitude but opposite direction then a = −b
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Vector representation
Types of vectors
(a) A position vector occurs when the point A is fixed
(b) A free vector is not restricted in any way. It is completely defined by its length and direction and can be drawn as any one of a set of equal length parallel lines
____AB
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Vector representation
Addition of vectors
The sum of two vectors and is defined as the single vector ____AB
____BC
____AC
____ ____ ____
or
AB BC AC
a b =c
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Vector representation
The sum of a number of vectors
Draw the vectors as a chain.
____ ____ ____ ____ ____
____or
AB BC CD DE AE
AE
a b c d
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Vector representation
The sum of a number of vectors
If the ends of the chain coincide the sum is 0.
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Components of a given vector
Just as can be replaced by so any single vector can be replaced by any number of component vectors so long as the form a chain beginning at P and ending at T.
____ ____ ____ ____AB BC CD DE
____AE
____PT
____PT a b c d
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Components of a given vector
Components of a vector in terms of unit vectors
The position vector , denoted by r can be defined by its two components in the Ox and Oy directions as:
____OP
(along O ) (along O )x y r a b
a b r i j
If we now define i and j to be unit vectors in the Ox and Oy directions respectively so that
then: and a b a i b j
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
In three dimensions a vector can be defined in terms of its components in the three spatial direction Ox, Oy and Oz as:
i is a unit vector in the Ox direction,j is a unit vector in the Oy direction and k is a unit vector in the Oz direction
Vectors in space
The magnitude of r can then be found from Pythagoras’ theorem to be:
2 2 2r a b c
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Direction cosines
The direction of a vector in three dimensions is determined by the angles which the vector makes with the three axes of reference:
so that
cos therefore cos
cos therefore cos
cos therefore cos
a b c
a a rrb b rrc c rr
r i j k
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Direction cosines
Since:
2 2 2 2
2 2 2 2 2 2 2
2 2 2 1
= then
cos cos cos
then
cos cos cos
a b c r
r r r r
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Direction cosines
Defining:
then:
where [l, m, n] are called the direction cosines.
cos
cos
cos
l
m
n
2 2 2 1l m n
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Scalar product of two vectors
If a and b are two vectors, the scalar product of a and b is defined to be the scalar (number):
where a and b are the magnitudes of the vectors and θ is the angle between them.
The scalar product (dot product) is denoted by:
cosab
cosab a.b
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Scalar product of two vectors
If a and b are two parallel vectors, the scalar product of a and b is then:
Therefore, given:
then:
1 2 3 1 2 3
1 1 2 2 3 3
and
a a a b b b
a b a b a b
a i j k b i j k
a.b
cos0ab ab a.b
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Vector product of two vectors
The vector product (cross product) of a and b, denoted by:
is a vector with magnitude:
and a direction perpendicular to both a and b such that a, b and form a right-handed set.
sinab
a b
a b
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Vector product of two vectors
If is a unit vector in the direction of:
then:
Notice that:
ˆsin ab a b n
a b
b a a b
n̂
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Vector product of two vectors
Since the coordinate vectors are mutually perpendicular:
and
i j kj k ik i j
i i j j k k 0
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Vector product of two vectors
So, given:
then:
That is:
1 2 3 1 2 3 and a a a b b b a i j k b i j k
2 3 3 2 1 3 3 1 1 2 2 1( ) ( ) ( )a b a b a b a b a b a b a b i j k
1 2 3
1 2 3
a a ab b b
i j k
a b
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Angle between two vectors
Let a have direction cosines [l, m, n] and b have direction cosines [l′, m′, n′]
Let and be unit vectors parallel to a and b respectively.
therefore
2 2 2 2( ) ( ) ( ) ( )2 2( )2 2cos by the cosine rule
PP l l m m n nll mm nn
cos =ll mm nn
____OP
____OP
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Introduction: scalar and vector quantities
Vector representation
Components of a given vector
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
STROUD
Worked examples and exercises are in the text
Programme 6: Vectors
Direction ratios
Since
the components a, b and c are proportional to the direction cosines they are sometimes referred to as the direction ratios of the vector.
and
, ,
a b c
a b cl m nr r r
r i j k
STROUD
Worked examples and exercises are in the text
Programme 6: VectorsLearning outcomes
Define a vector
Represent a vector by a directed straight line
Add vectors
Write a vector in terms of component vectors
Write a vector in terms of component unit vectors
Set up a system for representing vectors
Obtain the direction cosines of a vector
Calculate the scalar product of two vectors
Calculate the vector product of two vectors
Determine the angle between two vectors
Evaluate the direction ratios of a vector