If vector is multiplied by a positive scalar quantity m, the product is a vector that has the same direction as and magnitude m A. If vector is multiplied by a negative scalar quantity m, the product is directed opposite . For exam- ple, the vector is five times as long as and points in the same direction as ; the vector 13AS is one-third the length of and points in the direction opposite .AS AS AS AS
5AS
AS mAS
AS AS
mAS AS
ASBS BS
AS ASBS
ASBS AS 1BS2
AS BS
ASBS
AS AS
AS 1AS20
AS AS
Section 3.3 Some Properties of Vectors 57
B A B
C
(A B)
C
A
A B
B C C
A (B
C)
A
Figure 3.9 Geometric constructions for verifying the associative law of addition.
C A B
A B C A B
A
B B
(a) (b)
Figure 3.10 (a) This construction shows how to subtract vector from vector . The vector is equal in magnitude to vector and points in the opposite direction. To subtract from , apply the rule of vector addition to the combination of and : first draw along some convenient axis and then place the tail of at the tip of , and Cis the difference . (b) A second way of looking at vector subtraction. The difference vector CSASBSis the vector that we must add to to obtain .BS AS
ASBS A
BS S
A BS S
AS
A
S
BS
BS BS AS BS
48.2 km
R 2120.0 km22 135.0 km222120.0 km2 135.0 km2 cos 120°
Substitute numerical values, noting that u180° 60°120°:
Quick Quiz 3.2 The magnitudes of two vectors and are A12 units and B 8 units. Which of the following pairs of numbers represents the largest and smallest possible values for the magnitude of the resultant vector ? (a) 14.4 units, 4 units (b) 12 units, 8 units (c) 20 units, 4 units (d) none of these answers
Quick Quiz 3.3 If vector is added to vector , which two of the following choices must be true for the resultant vector to be equal to zero? (a) and are parallel and in the same direction. (b) and are parallel and in opposite direc- tions. (c) and AS BShave the same magnitude. (d) and AS BS are perpendicular.
BS AS
BS AS AS
BS
RS ASBS BS
AS
y (km) 40
20 60.0 R
A
x (km) 0
y (km)
B
20 A
x (km) 20 0
(b) N
S
W E
B
20
R
40
(a) b b u
Figure 3.11 (Example 3.2) (a) Graphical method for finding the resul- tant displacement vector . (b) Adding the vectors in reverse order 1BSAS2gives the same result for .RSASBS RS
E X A M P L E 3 . 2
A car travels 20.0 km due north and then 35.0 km in a direction 60.0° west of north as shown in Figure 3.11a. Find the magnitude and direction of the car’s resultant displacement.
SOLUTION
Conceptualize The vectors and drawn in Figure 3.11a help us conceptualize the problem.
Categorize We can categorize this example as a sim- ple analysis problem in vector addition. The displace- ment is the resultant when the two individual dis- placements and are added. We can further categorize it as a problem about the analysis of trian- gles, so we appeal to our expertise in geometry and trigonometry.
BS AS
RS
BS AS A Vacation Trip
Analyze In this example, we show two ways to analyze the problem of finding the resultant of two vectors. The first way is to solve the problem geometrically, using graph paper and a protractor to measure the magnitude of and its direction in Figure 3.11a. (In fact, even when you know you are going to be carrying out a calculation, you should sketch the vectors to check your results.) With an ordinary ruler and protractor, a large diagram typically gives answers to two-digit but not to three-digit precision.
The second way to solve the problem is to analyze it algebraically. The magnitude of can be obtained from the law of cosines as applied to the triangle (see Appendix B.4).
R
S
R
S
Use R2 A2 B2 2ABcos u from the law of cosines to find R:
R 2A2B22AB cos u
Use the law of sines (Appendix B.4) to find the direction of measured from the northerly direction:
RS
sin b B
R sin u 35.0 km
48.2 km sin 120°0.629 sin b
B sin u R
3.4 Components of a Vector and Unit Vectors
The graphical method of adding vectors is not recommended whenever high accu- racy is required or in three-dimensional problems. In this section, we describe a method of adding vectors that makes use of the projections of vectors along coor- dinate axes. These projections are called the components of the vector or its rec- tangular components. Any vector can be completely described by its components.
Consider a vector lying in the xyplane and making an arbitrary angle u with the positive x axis as shown in Figure 3.12a. This vector can be expressed as the sum of two other component vectors , which is parallel to the xaxis, and , which is parallel to the y axis. From Figure 3.12b, we see that the three vectors form a right triangle and that . We shall often refer to the “components of a vector ,” written Axand Ay(without the boldface notation). The component Ax represents the projection of along the x axis, and the component Ayrepresents the projection of along the y axis. These components can be positive or nega- tive. The component Axis positive if the component vector points in the posi- tive xdirection and is negative if points in the negative xdirection. The same is true for the component Ay.
From Figure 3.12 and the definition of sine and cosine, we see that cos u Ax/Aand that sin uAy/A. Hence, the components of are
(3.8) (3.9) AyA sin u
AxA cos u
AS ASx
A
S
x
AS
AS AS
A
SA
S
xA
S
y
ASy ASx
AS
Section 3.4 Components of a Vector and Unit Vectors 59
Finalize Does the angle b that we calculated agree with an estimate made by looking at Figure 3.11a or with an actual angle measured from the diagram using the graphical method? Is it reasonable that the magni- tude of is larger than that of both and ? Are the units of correct?
Although the graphical method of adding vectors works well, it suffers from two disadvantages. First, some
RS
BS AS RS
people find using the laws of cosines and sines to be awkward. Second, a triangle only results if you are adding two vectors. If you are adding three or more vec- tors, the resulting geometric shape is usually not a trian- gle. In Section 3.4, we explore a new method of adding vectors that will address both of these disadvantages.
y
x A
O
y
Ax (a)
y
O Ax x
(b)
Ay
A A
u u
Figure 3.12 (a) A vector lying in the xyplane can be represented by its component vectors and . (b) The ycomponent vector can be moved to the right so that it adds to . The vector sum of the component vectors is . These three vectors form a right triangle.AS
A
S
A x S y
ASy ASx A
S
Components of the vector AS PITFALL PREVENTION 3.2 Component Vectors versus Components
The vectors and are the com- ponent vectorsof . They should not be confused with the quantities Axand Ay, which we shall always refer to as the componentsof .A
S
AS A
S
A y S x
PITFALL PREVENTION 3.3 xand yComponents
Equations 3.8 and 3.9 associate the cosine of the angle with the xcom- ponent and the sine of the angle with the ycomponent. This associa- tion is true onlybecause we mea- sured the angle uwith respect to the xaxis, so do not memorize these equations. If uis measured with respect to the yaxis (as in some problems), these equations will be incorrect. Think about which side of the triangle contain- ing the components is adjacent to the angle and which side is oppo- site and then assign the cosine and sine accordingly.
What If? Suppose the trip were taken with the two vectors in reverse order: 35.0 km at 60.0° west of north first and then 20.0 km due north. How would the magnitude and the direction of the resultant vector change?
Answer They would not change. The commutative law for vector addition tells us that the order of vectors in an addition is irrelevant. Graphically, Figure 3.11b shows that the vectors added in the reverse order give us the same resultant vector.
The resultant displacement of the car is 48.2 km in a direction 38.9° west of north.
The magnitudes of these components are the lengths of the two sides of a right tri- angle with a hypotenuse of length A. Therefore, the magnitude and direction of are related to its components through the expressions
(3.10) (3.11) Notice that the signs of the components Ax and Ay depend on the angleu. For example, if u 120°, Axis negative and Ayis positive. If u 225°, both Axand Ay are negative. Figure 3.13 summarizes the signs of the components when lies in the various quadrants.
When solving problems, you can specify a vector either with its components Axand Ayor with its magnitude and direction Aand u.
Suppose you are working a physics problem that requires resolving a vector into its components. In many applications, it is convenient to express the components in a coordinate system having axes that are not horizontal and vertical but that are still perpendicular to each other. For example, we will consider the motion of objects sliding down inclined planes. For these examples, it is often convenient to orient the xaxis parallel to the plane and the yaxis perpendicular to the plane.
Quick Quiz 3.4 Choose the correct response to make the sentence true: A com- ponent of a vector is (a) always, (b) never, or (c) sometimes larger than the magni- tude of the vector.