Practice Problems: Vectors Solutions

1. (easy) Vector A represents 5.0 m of displacement east.  If vector B represents 10.0 m of displacement west, find the addition of the two displacements (Resultant vector = R).
R = 5.0 - 10.0 = -5.0 m
The resultant vector has a magnitude of 5.0 m and is west.

2. (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).
R = (5.0² + 10.0²)^1/2 =  11 m
tanθ = 10.0/5.0 = 2.0
θ = 63° 
R = 11m, 63° 

3. (easy) Determine the x and y components of a displacement whose magnitude is 30.0 m at a 23° angle from the x-axis.
x-comp = 30cos23 = 28 m
y-comp = 30sin23  = 12 m

4. (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.

5. (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.
F = 6N, east
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.
The forces are at right angles. Use the Pythagorean Theory.
F = (10² + 4²)^½ = 11 N
tanθ = 4/10
θ = 22º

6. (moderate) Find the equilibrant force for the system of forces described here:
Force A: 20 N at 20°
Force B: 40 N at 230°

7. (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 displacents? 22 m, 2 m, 30.9 m, 15.6 m. For the possible resultants, what angle exists between the original displacements?

If the displacements are parallel (0° angle between them): R = 22 m
If the displacements are antiparallel (180° angle between them): R = 2 m
If the displacements are perpendicular (90° angle between them): R = 15.6 m
(Note: Use the Law of Cosines to obtain the angle for R = 15.6 m)
Resultants greater than 22 m are not possible.

8. (moderate) Find the dot product (A • B) and the cross product magnitude |A x B| for these two vectors:
Vector A: magnitude of 15 units, angle of 10°
Vector B: magnitude of 150 units, angle of 80°
A • B = ABcosθ = 15(150)cos(70) = 770 units²
|A x B| = ABsinθ = 15(150)sin(70) = 2100 units²

9. (moderate)If vector C has a magnitude 100 units pointing along the x-axis, and Vector D has a magnitude of 50 units, at what angle must it point so that the dot product of Vector C and Vector D has a magnitude of 250 units?
A • B = ABcosθ = 250 = 100(50)cosθ
cosθ = 0.05
θ = 87°

10. (moderate) A student carries a lump of clay from the ground floor 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.