1. A car is driven east for a distance of 42 km, then north for 25 km, and then in a direction 30° east of north for 28 km. Determine (a) the magnitude of the car’s total displacement from its starting point and (b) the angle (from east) of the car’s total displacement measured from its starting direction.
a. 74.58 Units: km
b. 41.33 Units: Degrees
2. For the vectors a = 3.0i + 4.0j and b = 5.0i + (-2.0j), give vector a + vector b in (a) unit-vector notation, and as (b) a magnitude and (c) an angle (relative to i in the range of (-180°, 180°]). Now give in (d) unit-vector notation, and as (e) a magnitude and (f) an angle (relative to in the range of (-180°, 180°])
a. 8.0i + 2.0j Units: m
b. 8.25 Units: m
c. 14.04 Units: degrees
d. 2.0i + (-6.0j) Units: m
3. Two vectors are given by a = 8.0i+5.4j and b = 1.9i + 8.1j. Find (a) |a x b|, (b) a*b, (c) (a + b)*b, and (d) the component of a along the direction of b?
a. 54.54 Units: No Units
b. 58.94 Units: No Units
c. 128.16 Units: No Units
d. 7.08 Units: No Units
4. Two vectors, and lie in the xy plane. Their magnitudes are 3.39 and 5.69 units, respectively, and their directions are 317o and 51o, respectively, as measured counterclockwise from the positive x axis. What are the values of (a) and (b) ?
a. -1.33 Units: No Units
b. 19.23 Units: No Units
Answer: Vector B = 3.0i + 3.0j + (-3.0k)
7. Rock faults are ruptures along which opposite faces of rock have slid past each other. In the figure, points A and B coincided before the rock in the foreground slid down to the right. The net displacement is along the plane of the fault. The horizontal component of is the strike-slip AC. The component of that is directly down the plane of the fault is thedip-slip AD. (a) What is the magnitude of the net displacement if the strike-slip is 21.8 m and the dip-slip is 17.5 m? (b) If the plane of the fault is inclined at angle = 56 ° to the horizontal, what is the magnitude of the vertical component of ?
a. 27.96 Units: m
b. 14.5 Units: m
8. Find the sum of the following four vectors ((a) and (b) for x and y components respectively), and as (c) a magnitude and (d) an angle relative to +x (including sign plus or minus).
: 17 m, at 17o counterclockwise from +x
: 22 m, at 10o counterclockwise from +y
: 23 m, at 34o clockwise from -y
: 9.0 m, at 48o counterclockwise from –y
a. 6.27 Units: m
b. 1.55 Units: m
c. 6.46 Units: m
d. 13.89 Units: m
9. A bank in downtown Boston is robbed (see the map in the figure). To elude police, the robbers escape by helicopter, making three successive flights described by the following displacements: 32 km, 45o south of east; 53 km, 26o north of west; 26 km, 18o east of south. At the end of the third flight they are captured. In what town are they apprehended?
11. The position of a particle moving in an xy plane is given by with in meters and t in seconds. In unit-vector notation, calculate
(a) , (b) , and (c) for t = 2 s. (d) What is the angle between the positive direction of the x axis and a line tangent to the particle’s path at t = 2 s? Give your answer in the range of (-180o; 180o).
12. In the figure, a radar station detects an airplane approaching directly from the east. At first observation, the airplane is at distance d1 = 390 m from the station and at angle θ1 = 43°above the horizon. The airplane is tracked through an angular change Δθ = 115° in the vertical east–west plane; its distance is then d2 = 790 m. Find the (a) magnitude and (b)direction of the airplane’s displacement during this period. Give the direction as an angle relative to due west, with a positive angle being above the horizon and a negative angle being below the horizon.
a. 1018.144 Units: m
b. 1.68622 Units: degrees