Transcript
Vectors and ScalarsWeek 2
PHYSICS 10 - General Physics 1
Objectives•Differentiate vectors and scalar quantities•Add vectors using graphical methods•Determine the components of a vector•Determine the magnitude and direction of
vectors using its components•Define unit vectors
Vector and Scalar
•Scalar quantity – described by a single number
•Vector quantity – has both magnitude and direction in space
•Examples:▫Temperature = 20°C (Scalar)▫Displacement = 20 m, South (Vector)
Vectors•Represented by a letter or symbol in boldface
italic type with an arrow above them
•Arrows are used to represent vectors geometrically, plotted in a Cartesian plane
•Two vectors are parallel if they same direction (otherwise they are antiparallel)
•The magnitude of a vector, is a scalar quantity
Vector addition and subtraction•Graphical methods
▫Parallelogram (or triangle law)
▫Polygon method
Vector addition and subtraction
•Resultant vector – vector sum
•Properties▫Commutative: ▫Associative:
Vector addition and subtraction• Subtracting vectors
(negating a vector is equivalent to changing its direction to the opposite)
Thus:
Review Questions1. A circular racetrack has a radius of 500 m. What is
the displacement of a cyclist when she travels around the track from the north side to the south side? When she makes one complete circle around the track? Explain your reasoning.
2. Can you find two vectors with different lengths that have a vector sum of zero? What length restrictions are required for three vectors to have a vector sum of zero? Explain your reasoning.
3. Hearing rattles from a snake, you make two rapid displacements of magnitude 1.8 m and 2.4 m. In sketches (roughly to scale), show how your two displacements might add up to give a resultant of magnitude (a) 4.2 m; (b) 0.6 m; (c) 3.0 m.
Components of Vectors
• Finding a vector’s magnitude and direction from its components.
and
• Using components to calculate the vector sum (resultant) of two or more vectors.
Components of Vectors
Exercise1. Find the magnitude and direction of the vector
represented by the following pairs of components: (a) , (b) ,
2. Vector is 2.80 cm long and is above the x-axis in the first quadrant. Vector is 1.90 cm long and is below the x-axis in the fourth quadrant. Use components to find the magnitude and direction of (a) (b) (c) . In each case, sketch the vector addition or subtraction and show that your numerical answers are in qualitative agreement with your sketch.
Unit Vectors
Unit Vectors
Position and position vector• We define position as the location of an object
in space and specified in a coordinate axis
• In determining the position of an object in space, we use the position vector,
• The position vector determines the location of the object in space
r
jrirr yxˆˆ
PositionExample:The location of the hen is
Where 8 m is the x component of the hen’s position and 4 is the vertical component of the hen’s position
5r
m j4i8 r
Displacement • Displacement is the shortest distance between two points
• Displacement is defined as the change in the position between two points
• Displacement is a vector quantity
if rrr
DisplacementExample:Determine the displacement of an object that is initially located at and final position at Solution:
mˆ0.8ˆ0.2 jiri mˆ0.3ˆ0.4 jirf
m ˆ0.8ˆ0.2m ˆ0.3ˆ0.4 jijir if rrr
mjir ˆ0.11ˆ0.2
Exercise1. (a) Write each vector in the figure in
terms of the unit vectors and and find 2. (b) Find the magnitude and direction of .
Problems1. Three horizontal ropes pull on a large stone stuck in the ground, producing the vector forces , and shown in the fig. Find the magnitude and direction of a fourth force on the stone that will make the vector sum of the four forces zero.
Problems2. A sailor in a small sailboat encounters shifting winds. She sails 2.00 km east, then 3.50 km southeast, and then an additional distance in an unknown direction. Her final position is 5.80 km directly east of the starting point (Fig. P1.72). Find the magnitude and direction of the third leg of the journey. Draw the vector addition diagram and show that it is in qualitative agreement with your numerical solution.
Review
• When two vectors and represented in terms of their components, we express vector sum using unit vectors:
A
RB
)15.1(ˆˆ
ˆˆ
ˆˆˆˆ
ˆˆ
ˆˆ
jRiR
jBAiBA
jBiBjAiA
BAR
jBiBB
jAiAA
yx
yyxx
yxyx
yx
yx
• If vectors do not all lie in the xy-plane, we need a third component. We introduce a third unit vector that points in the direction of the positive z-axis.
k
)17.1(ˆˆˆ
ˆˆˆ
)16.1(ˆˆˆ
ˆˆˆ
kRjRiRR
kBAjBAiBAR
kBjBiBB
kAjAiAA
zyx
zxyyxx
zyx
zyx
Example 1.9 Using Unit Vectors
Given the two displacements
Find the magnitude of the displacement .
Solution:Identify, Set Up and Execute:Letting , we have
mandm kjiEkjiD ˆ8ˆ5ˆ4ˆˆ3ˆ6
ED
2
m
m
m
kjiF
kji
kjikjiF
ˆ10ˆ11ˆ8
ˆ82ˆ56ˆ412
ˆ8ˆ5ˆ4ˆˆ3ˆ62
EDF
2
Example 1.9 (SOLN)
The units of the vectors , , and are meters, so the components of these vectors are also in meters. From Eqn 1.12,
Evaluate:Working with unit vectors makes vector addition and subtraction no more complicated than adding or subtraction ordinary numbers. Be sure to check for simple arithmetic errors.
D
mmmm 1710118 222
222
F
FFFF zyx
E
F
Products of VectorsKinds:1. Scalar Product (dot product) ●results to a scalar quantity
2. Vector Product (cross-product)
● yields to another vector
Right-hand Rule
Vector product
B
BA
Review of Products of Vectors
• We can express many physical relationships concisely by using products of vectors.
• There are two different kinds of products of vectors. The first, called the scalar product (dot product), which yields a result that is a scalar quantity. The second is the vector product (cross product) which yields another vector.
Scalar product• The scalar product of two vectors and is
denoted by . • The scalar product is also called the dot
product.
A
B
BA
Scalar product• We define to be the magnitude of
multiplied by the component of parallel to ,
BA
A
B
A
)18.1(coscos BAABBA
Scalar product• We can also express .• The scalar product of two perpendicular
vectors is always zero.
Scalar product0° to 90° Positive
90° Zero90° to 180° Negative
coscos ABABBA
Scalar product• Scalar product obeys the commutative law of
multiplication; order of the two vectors does not matter.
• We can calculate the scalar product directly if we know the x-, y-, and z-components of and . Using Eqn 1.18, we find
BA
A
B
)19.1(90cos11ˆˆˆˆˆˆ
10cos11ˆˆˆˆˆˆ
kjkiji
kkjjii
1.10 Products of Vectors
Scalar product• Now, we express and in terms of their
components, expand the product, and use the products of unit vectors
A
B
)20.1(ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
kkBAjkBAikBA
kjBAjjBAijBA
kiBAjiBAiiBA
kBjBiBkAjAiABA
zzyzxz
zyyyxy
zxyxxx
zyxzyx
1.10 Products of Vectors
Scalar product• From Eqns 1.19, six of the nine terms are
zero, thus
• Scalar product of two vectors is the sum of the products of their respective components.
• Eqn 1.21 can also be used to find the scalar product of and .
21.1zzyyxx BABABABA
A
B
Sample problem : scalar product
Find the scalar product of the two vectors shown in figure. The magnitudes of the vectors are A = 4.00 and B = 5.00.
BA
(SOLUTION)
Identify and Set Up:2 ways to calculate the scalar product. First way uses the magnitudes of the vectors and the angle between them (Eqn 1.18), and the second uses the components of the two vectors (Eqn 1.21).Execute:Using first approach, angle between the 2 vectors is =130.0 53.0 = 77.0 , so
This is positive as angle between and is between 0 and 90 .
50.40.77cos00.500.4cos ABBA
A
B
Execute:For second approach, find the components of the 2 vectors. Since angles of and are given with respect to the +x-axis, and these angles are measured in the sense from the +x-axis to the +y-axis, we use Eqns 1.17:
0
830.30.130sin00.5
214.30.130cos00.5
0
195.30.53sin00.4
407.20.53cos00.4
z
y
x
x
y
x
B
B
B
A
A
A
A
B
Example 1.10 (SOLN)
Execute:The z-components are zero because both vectors lie in the xy-plane. As in Example 1.7, we are keeping one too many significant figures in the components and will round them at the end. From Eqn 1.21,
Evaluate:We get the same result for the scalar product with both methods, as we should.
50.4
00830.3195.3214.3407.2
zzyyxx BABABABA
Key Equations
)10.1(yyy
xxx
BAR
BAR
)16.1(ˆˆˆ
ˆˆˆ
kBjBiBB
kAjAiAA
zyx
zyx
)22.1(sinABC
27.1xyyxz
zxxzy
yzzyx
BABAC
BABAC
BABAC
)18.1(coscos BAABBA
21.1zzyyxx BABABABA
Finding Angles with Scalar Product
Find the angle between the two vectors
Solution:Identify:The scalar product of two vectors and is related to the angle between them and to the magnitudes A and B. The scalar product is also related to the components of the two vectors. If we are given the components of the vectors, we first determine the scalar product and the values of A and B, then determine the target variable .
kjiBkjiA ˆˆ2ˆ4ˆˆ3ˆ2
and
BA
A
B
Finding Angles with Scalar Product
Set Up and Execute:Use either Eqns 1.18 or 1.21. Equating these two and rearranging,
This formula can be used to find the angle between any two vectors and . The components of are Ax = 2, Ay = 3 and Az = 1, and components of are Bx = 4, By = 2 and Bz = 1.
AB
BABABA zzyyxx cos A
B
A
B
A
B
Finding Angles with Scalar Product
Set Up and Execute:Thus,
100
175.02114
3cos
21124
14132
3112342
222222
222222
AB
BABABA
BBBB
AAAA
BABABABA
zzyyxx
zyx
zyx
zzyyxx
Example 1.11 Finding Angles with Scalar Product
Evaluate:As a check on this result, note that the scalar product is negative. This means that is between 90 and 180 , in agreement with our answer.
BA
1.10 Products of Vectors
Vector product• Vector product of two vectors and , also
called the cross product, denoted by .• To define the vector product, we draw the
vectors as shown.
A
B
BA
1.10 Products of Vectors
Vector product• The vector product is defined as a vector
quantity with a direction perpendicular to the plane (both and ) and a magnitude equal to AB sin .
• If
• The vector product of any two parallel or antiparallel vectors is always zero.
• The vector product of any vector with itself is zero.
A
B
BAC
)22.1(sinABC
1.10 Products of Vectors
Vector product• There are always two directions
perpendicular to a given plane, one on each side of the plane. We choose which of these is the direction of .
• This right-hand rule is what we use to determine the direction of the vector product.
• Note that vector product is not commutative. In fact, for any two vectors and
BA
)23.1(ABBA
A
B
1.10 Products of Vectors
Vector product• If we know the components of and , we
can calculate the components of the vector product, using a procedure similar to scalar product.
• First, we work out the multiplication table for the unit vectors The vector product of any vector with itself is zero, so
• The boldface zero illustrates that each product is a zero vector, that is, all components equal to zero with an undefined direction.
.ˆ,ˆ,ˆ kji
A
B
0 kkjjii ˆˆˆˆˆˆ
1.10 Products of Vectors
Vector product• Using Eqns 1.22 and 1.23 and the right-
hand rule, we find
24.1ˆˆˆˆˆ
ˆˆˆˆˆ
ˆˆˆˆˆ
jkiik
ijkkj
kijji
1.10 Products of Vectors
Vector product• Express and in terms of their
components and corresponding unit vectors,
A
B
25.1ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
)ˆˆˆ()ˆˆˆ(
kBkAjBkAiBkA
kBjAjBjAiBjA
kBiAjBiAiBiA
kBjBiBkAjAiABA
zzyzxz
zyyyxy
zxyxxx
zyxzyx
1.10 Products of Vectors
Vector product• Rewrite individual terms as
and so on. Evaluating with the multiplication table for the unit vectors and grouping the terms, we find
• Thus the components of are given by
27.1xyyxz
zxxzyyzzyx
BABAC
BABACBABAC
)26.1(ˆ
ˆˆ
kBABA
jBABAiBABABA
zyyx
zxxzyzzy
jiBAjBiA yxyx ˆˆˆˆ
BAC
1.10 Products of Vectors
Vector product• Vector product can also be expressed in
determinant form as:
zyx
zyx
BBB
AAA
kji
BA
ˆˆˆ
1.10 Products of Vectors
Vector product• All vector products of the unit vectors
would have signs opposite to those in Eqn 1.24.
• Axis system where is called a right-handed system. The usual practice is to use only right-handed systems.
kji ˆ,ˆ, and
kji ˆˆˆ
Example 1.12 Calculating a vector product
Vector has magnitude 6 units and is in the direction of the +x-axis. Vector has magnitude 4 units and lies in the xy-plane, making an angle of 30 with the +x-axis. Find the vector product .
A
B
BA
Example 1.12 (SOLN)
Identify and Set Up:Find the vector product in one of two ways. First way is to use Eqn 1.22 to determine the magnitude ofand then use the right-hand rule to find the direction of the vector product. Second way is to use the components of and to find the components of the vector product using Eqn 1.27.Execute:With first approach using Eqn 1.22,
BA
A
B
BAC
1230sin46sin AB
Example 1.12 (SOLN)
Execute:From right-hand rule, direction of is along the +z-axis, so we have .For second approach, we first write the components of
Defining , we have
030sin43230cos4
006
zyx
zyx
BBB
AAA
kBA ˆ12 BA
BA
and
BAC
1232026
006320
02000
z
y
x
C
C
C
Example 1.12 (SOLN)
Execute:The vector product has only a z-component, and it lies along the +z-axis. The magnitude agrees with the result we obtained with the first approach, as it should.Evaluate:The first approach was more direct because the magnitudes of each vector and angle between them was known, and both vectors lay in one of the planes of the coordinate system. Sometimes, we need to find the vector product of 2 vectors that are not conveniently oriented or which only components are given. We use the second approach for such cases.
C
Concepts Summary
• Fundamental physical quantities of mechanics are mass, length and time.
• Corresponding basic SI units are the kilogram, the meter, and the second.
• Other units for these quantities, related by powers of 10, are identified by adding prefixes to the basic units.
• Derived units for other physical quantities are products or quotients of the basic units.
• Equations must by dimensionally consistent; two terms can be added only when they have the same units.
Concepts Summary
• Accuracy of a measurement can be indicated by the number of significant figures or by a stated uncertainty.
• Result of a calculation usually has no more significant figures than the input data.
• When only crude estimates are available for input data, we can often make useful order-of-magnitude estimates.
• Scalar quantities are numbers, and combine with usual rules of arithmetic.
Concepts Summary
• Vector quantities have direction as well as magnitude, and combine according to the rules of vector addition.
• Graphically, two vectors and are added by placing the tail of at the head, or tip, of .
• The vector sum then extends from the tail of to the head of .
• Vector addition can be carried out using components of vectors.
• The x-component of is the sum of the x-components of and , and likewise for y- and z-components.
A
B
BA
BAR
B
A
A
B
A
B
Concepts Summary
• Unit vectors describe directions in space.• A unit vector has a magnitude of one, with no
units.• The unit vectors, aligned with the x-, y-,
and z-axes of a rectangular coordinate system, are especially useful.
• The scalar product of two vectors and is a scalar quantity .
• It can be expressed in two ways: in terms of the magnitudes of and and the angle between the two vectors, or in terms of the components of and .
BAC
kji ˆ,ˆ,ˆ
A
B
A
B
A
B
Concepts Summary
• The scalar product is commutative; for any two vectors and , .
• The scalar product of two perpendicular vectors is zero.
• The vector product of two vectors and is another vector .
• The magnitude of depends on the magnitudes of and and the angle between the two vectors.
• The direction of is perpendicular to the plane of the two vectors being multiplied, as given by the right-hand rule.
ABBA
BAC
C
BA
A
B
A
B
A
B
BA
Concepts Summary
• The components of can be expressed in terms of the components of and .
• The vector product is not commutative; for any two vectors and , .
• The vector product of two parallel or antiparallel vectors is zero.
BAC
ABBA
A
B
A
B
Key Equations
)10.1(yyy
xxx
BAR
BAR
)16.1(ˆˆˆ
ˆˆˆ
kBjBiBB
kAjAiAA
zyx
zyx
)22.1(sinABC
27.1xyyxz
zxxzy
yzzyx
BABAC
BABAC
BABAC
)18.1(coscos BAABBA
21.1zzyyxx BABABABA
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