Practice Problems: Vectors
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1. (easy) Vector A represents 5.0 m of displacement east. If vector B represents 10.0 m of displacement north, find the addition of the two displacements (R).
2. (easy) Determine the x and y components of a displacement whose magnitude is 30.0 m at a 23° angle from the x-axis.
3. (moderate) A car moves 150.0 m at a 63° "north of east" (this simply means 63° from the x-axis). It stays at rest for a while then moves 300 m at 34° "south of west" (this means 214° from the x-axis.) Find the total displacement of the car.
4. (easy) Two forces are being exerted on an object, but in different directions. For example, you and a friend might both be pulling on strings attached to a single block of wood. Find the magnitude and direction of the resultant force in the following circumstances.
a) The first force has a magnitude of 10 N and acts east. The second force has a magnitude of 4 N and acts west.
b) The first force has a magnitude of 10 N and acts east. The second force has a magnitude of 4 N and acts north.
5. (moderate) Find the equilibrant force for the system of forces described here:
Force A: 20 N at 20°
Force B: 40 N at 230°
6. (moderate) Vector A represents a displacement in meters expressed in unit vector notation as
A = 2i + 6j + 3k
Vector B represents a second displacement.
B = 5i -3j – 2k
Find the dot product of the two vectors, the cross product of the two vectors, and the angle between them.
7. (moderate) Vector D = 3i - 4j + 2k and vector: E = 4i - j - 2k. Find the magnitude of D + E and the magnitude of D - E.
8. (moderate) If force vector F1 has a magnitude of 30 N pointing in the -z direction and force vector F2 has a magnitude of 60 N pointing in the +x direction, determine the dot product (F1 • F2) and the cross product (F1 x F2). How would the answers change if the vectors switched position in the equations?
9. (moderate) Two displacements with magnitudes of 10 m and 12 m can be combined to form resultant vectors with many different magnitudes. Which of the following magnitudes can result from these two displacments? 22 m, 2 m, 30.9 m, 15.6 m. For the possible resultants, what angle exists between the original displacements?
10. (moderate) A bicycle tire (Radius = R = 0.4 m) rolls along the ground (with no slipping) through three-quarters of a revolution. Consider the point on the tire that was originally touching the ground. How far has it displaced from its starting position?
11. (moderate) A student carries a lump of clay from the first floor (ground level) door of a skyscraper (on Grant Street) to the elevator, 24 m away. She then takes the elevator to the 11th floor. Finally, she exits the elevator and carries the clay 12 m back toward Grant Street. Determine the total displacement for the clay if each floor is 4.2 m above the floor below.
Please supplement these problems with those found in your companion text.