**Practice Problems: ****Vectors**Click here to see the solutions.

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** = 2**i** + 6**j** + 3**k**

Vector **B** represents a second displacement.

**B** = 5**i** -3**j** – 2**k**

Find the dot product of the two vectors, the cross product of the two vectors, and the angle between them.

7. (moderate) Vector **D** = 3**i** - 4**j** + 2**k** and vector: **E** = 4**i** - **j** - 2**k**.** **Find** **the magnitude of **D** + **E** and the magnitude of **D** - **E**.

8. (moderate) If force vector** F _{1}** has a magnitude of 30 N pointing in the -z direction and force vector

**F**has a magnitude of 60 N pointing in the +x direction, determine the dot product (

_{2}**F**•

_{1}**F**) and the cross product (

_{2}**F**x

_{1}**F**). How would the answers change if the vectors switched position in the equations?

_{2}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.**