is very different from ABC. The first is a vector sum, which must be handled graphically or with components, while the second is a simple arithmetic sum of numbers.
C:
B:
A:
(b)
42 Chapter 3 Vectors and Two-Dimensional Motion
When two vectors are added, their sum is independent of the order of the addition: . This relationship can be seen from the geometric construction in Active Figure 3.2b, and is called the commutative law of addition.
This same general approach can also be used to add more than two vectors, as is done in Figure 3.3 for four vectors. The resultant vector sum is the vector drawn from the tail of the first vector to the tip of the last. Again, the order in which the vectors are added is unimportant.
Negative of a Vector. The negative of the vector is defined as the vector that gives zero when added to . This means that and have the same magnitude but opposite directions.
Subtracting Vectors. Vector subtraction makes use of the definition of the nega- tive of a vector. We define the operation as the vector added to the vec- tor :
( ) [3.1]
Vector subtraction is really a special case of vector addition. The geometric con- struction for subtracting two vectors is shown in Figure 3.4.
Multiplying or Dividing a Vector by a Scalar. Multiplying or dividing a vector by a scalar gives a vector. For example, if vector is multiplied by the scalar number 3, the result, written 3 , is a vector with a magnitude three times that of and point- ing in the same direction. If we multiply vector by the scalar 3, the result is3 , a vector with a magnitude three times that of and pointing in the op- posite direction (because of the negative sign).
:A
:A :A
:A
A: A:
:B A:
B: :A
:A A: B: B:
A:
A:
A: A:
D: R: A: :B C:
:A B:
B:
A:
(a) R = A + B
A
A B
B
R = A + B
ACTIVE FIGURE 3.2
(a) When vector is added to vector , the vector sum is the vector that runs from the tail of to the tip of . (b) Here the resultant runs from the tail of to the tip of . These con- structions prove that
Log into PhysicsNow at http://
physics.brookscole.com/ecpand go to Active Figure 3.2 to vary and and see the effect on the resultant.
B
: A:
A
:
. B
:
B
:
A A:
B : :
B:
A:
R
:
A
: :B
The magnitudes of two vectors and are 12 units and 8 units, respectively. What are the largest and smallest possible values for the magnitude of the resultant vector R: A:B:? (a) 14.4 and 4; (b) 12 and 8; (c) 20 and 4; (d) none of these.
:B A:
Quick Quiz 3.1
If vector is added to vector , the resultant vector has magnitude AB when and are (a) perpendicular to each other; (b) oriented in the same direc- tion; (c) oriented in opposite directions; (d) none of these answers.
:B
A: B: :A A: B:
Quick Quiz 3.2
A – B – B
A
B
Figure 3.4 This construction shows how to subtract vector from vector . The vector has the same magnitude as the vector , but points in the opposite direction.
B B :
A :
: B:
R = A + B + C + D A
C
B D
Figure 3.3 A geometric construc- tion for summing four vectors.
The resultant vector is the vector that completes the polygon.
R
:
3.2 Components of a Vector 43
EXAMPLE 3.1 Taking a Trip
Goal Find the sum of two vectors by using a graph.
Problem A car travels 20.0 km due north and then 35.0 km in a direction 60west of north, as in Figure 3.5. Using a graph, find the magnitude and direction of a sin- gle vector that gives the net effect of the car’s trip. This vector is called the car’s re- sultant displacement.
Strategy Draw a graph, and represent the displacement vectors as arrows. Graphi- cally locate the vector resulting from the sum of the two displacement vectors. Mea- sure its length and angle with respect to the vertical.
Solution
Let represent the first displacement vector, 20.0 km north, the second displace- ment vector, extending west of north. Carefully graph the two vectors, drawing a resultant vector with its base touching the base of and extending to the tip of . Measure the length of this vector, which turns out to be about 48 km. The angle , measured with a protractor, is about 39west of north.
Remarks Notice that ordinary arithmetic doesn’t work here: the correct answer of 48 km is not equal to 20.0 km35.0 km55.0 km!
Exercise 3.1
Graphically determine the magnitude and direction of the displacement if a man walks 30.0 km 45north of east and then walks due east 20.0 km.
Answer 46 km, 27north of east
B: :A
R:
B:
A:
y(km)
40
20 60.0°
x(km) 0
β N
S
W E
B
–20 R
A
Figure 3.5 (Example 3.1) A graph- ical method for finding the resultant displacement vector R:A::B.
3.2 COMPONENTS OF A VECTOR
One method of adding vectors makes use of the projections of a vector along the axes of a rectangular coordinate system. These projections are called components.
Any vector can be completely described by its components.
Consider a vector in a rectangular coordinate system, as shown in Figure 3.6.
can be expressed as the sum of two vectors: x, parallel to the x-axis; and y, parallel to the y- axis. Mathematically,
x y
where xand yare the component vectors of . The projection of along the x-axis, Ax, is called the x- component of , and the projection of along the y-axis, Ay, is called the y- component of . These components can be either posi- tive or negative numbers with units. From the definitions of sine and cosine, we see that cos Ax/Aand sin Ay/A, so the components of are
AxAcos
[3.2]
AyAsin
These components form two sides of a right triangle having a hypotenuse with magnitude A. It follows that ’s magnitude and direction are related to its compo- nents through the Pythagorean theorem and the definition of the tangent:
A:
A:
A::A A:
:A
:A
:A A:
A: :A
:A
A:
A:
A: :A
y
O x
tan =Ay Ax
Ay
Ax θ
θ A
Figure 3.6 Any vector lying in the xy-plane can be represented by its rectangular components Axand Ay.
A
:
TIP 3.2 x- and y-Components
Equation 3.2 for the x- and y-compo- nents of a vector associates cosine with the x-component and sine with the y- component, as in Figure 3.7a.
(See page 44.) This association is due solelyto the fact that we chose to measure the angle with respect to the positive x-axis. If the angle were measured with respect to the y-axis, as in Figure 3.7b, the components would be given by AxAsin and AyAcos .
[3.3]
[3.4]
To solve for the angle , which is measured from the positive x-axis by convention, we can write Equation 3.4 in the form
tan1
This formula gives the right answer only half the time! The inverse tangent func- tion returns values only from 90to 90, so the answer in your calculator win- dow will only be correct if the vector happens to lie in the first or fourth quadrant.
If it lies in the second or third quadrant, adding 180to the number in the calcu- lator window will always give the right answer. The angle in Equations 3.2 and 3.4 must be measured from the positive x-axis. Other choices of reference line are possible, but certain adjustments must then be made. (See Tip 3.2 [page 43] and Figure 3.7.)
If a coordinate system other than the one shown in Figure 3.6 is chosen, the components of the vector must be modified accordingly. In many applications it’s more convenient to express the components of a vector in a coordinate system having axes that are not horizontal and vertical, but are still perpendicular to each other. Suppose a vector makes an angle with the x-axis defined in Figure 3.8.
The rectangular components of along the axes of the figure are given by BxB cos and ByBsin , as in Equations 3.2. The magnitude and direc- tion of :Bare then obtained from expressions equivalent to Equations 3.3 and 3.4.
:B B:
AAxy
tan Ay Ax
A√Ax2 Ay2
44 Chapter 3 Vectors and Two-Dimensional Motion
y
0 x
Ay = A sin θ
θ
Ax = A cosθ A
(a) y
0 x Ay = A cos
θ θ
Ax = A sinθ
A
(b)
Figure 3.7 The angle need not always be defined from the positive x-axis.
x′
y′
′ B
By′
Bx′
O′ θ
Figure 3.8 The components of vector in a tilted coordi- nate system.
B
:
EXAMPLE 3.2 Help Is on the Way!
Goal Find vector components, given a magnitude and direction, and vice versa.