Solutions of Homework Problems Vectors in Physics
Solutions of Homework Problems Vectors in Physics
Solutions of Homework Problems Vectors in Physics
12. Picture the Problem: The given vector components correspond to the vector r as drawn at y
right.
14 m
x
9.5m
(a) Use the inverse tangent function to find the tan 1
34 or 34° below −9.5 m
distance angle : 14 m
r
the +x axis
14 m 9.5 m
2 2
(b) Use the Pythagorean Theorem to r rx2 ry2
determine the magnitude of r :
r 17 m
9.5m 2
(c) If both rx and ry are doubled, the direction tan 1 34
14 m 2
will remain the same but the magnitude will
28 m 19 m 34 m
2 2
double: r
15. Picture the Problem: The two vectors A (length 50 units) and B (length 120 units) are drawn y
at right.
A
x
Solution: 1. (a) Find Bx: Bx 120 units cos 70 41 units 70°
2. Since the vector A points entirely in the x direction, we can see that Ax = 50 units and that
vector A has the greater x component.
B
3. (b) Find By: Bx 120 units sin 70 113 units
4. The vector A has no y component, so it is clear that vector B has the greater y component. However, if one takes
into account that the y-component of B is negative, then it follows that it smaller than zero, and hence A has the
greater y-component.
20. The two vectors A (length 40.0 m) and B (length 75.0 m) are drawn at right. y
B A
(a) A sketch (not to scale) of the vectors and their sum is shown at right.
50.0° C
x
20.0°
(b) Add the x components: C x Ax Bx 40.0 m cos 20.0 75.0 m cos 50.0 85.8 m
Add the y components: C y Ay B y 40.0 m sin 20.0 75.0 m sin 50.0 43.8 m
85.8 m 43.8 m 96.3 m
2 2
Find the magnitude of C : C Cx C y
2 2
Cy 1 43.8 m
Find the direction of C : C tan 1 tan 27.0
Cx 85.8 m
3–1
24. The vectors involved in the problem are depicted at right. y
Set the length of A B equal 37 A2 B 2 30
2 2 2
to 37 units: 37 A B AB
B 37 2 A2 372 22 30 units B
2
Solve for B:
x
−22 A O
29. The vector A has a length of 6.1 m and points in the negative x direction.
Note that in order to multiply a vector by a scalar, you need only multiply each component of the vector by the same
scalar.
(a) Multiply each component of A by −3.7: A 6.1 m xˆ
3.7 A 3.7 6.1 m xˆ 23 m xˆ so Ax 23 m
(b) Since A has only one component, its magnitude is simply 23 m.
31. Picture the Problem: The vectors involved in the problem are depicted at right. y
2.0 m
(a) Find the direction of A from its A tan 1 –22
components: 5.0 m AB
B
5.0 m 2.0 m 5.4 m
2 2
Find the magnitude of A : A A B
2.0 m 5.0 m x
O
2.0 m A
5.0 m
(b) Find the direction of B from its B tan 1 68 180 110
components: –2.0 m
2.0 m 5.0 m 5.4 m
2 2
Find the magnitude of B : B
(c) Find the components of A B : A B 5.0 2.0 m xˆ 2.0 5.0 m yˆ 3.0 m xˆ 3.0 m yˆ
3.0 m
Find the direction of A B from its A B tan 1 45
components: 3.0 m
Find the magnitude of A B : AB 3.0 m 2 3.0 m 2 4.2 m
3–2