11.4 The Cross Product of Two Vectors in Space 671
11.4 The Cross Product of Two Vectors in Space
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the cross product of the unit vector .
a.
b.
c.
d.
e.
____ 2. Sketch the graph of the cross product of the unit vector .
a. d.
b. e.
672 Chapter 11: Vectors and the Geometry of Space
c.
____ 3. Given , find .
a.
b.
c.
d.
e.
____ 4. Given and find .
a.
b.
c.
d.
e.
____ 5. Given and find .
a.
b.
c.
d.
e.
11.4 The Cross Product of Two Vectors in Space 673
____ 6. Given and , find .
a.
b.
c.
d.
e.
____ 7. Given and , find .
a.
b.
c.
d.
e.
____ 8. Find a unit vector that is orthogonal to both and .
a.
b.
c.
d.
e.
____ 9. Find the area of the parallelogram that has the given vectors and as
adjacent sides.
a.
b.
c.
d.
e.
674 Chapter 11: Vectors and the Geometry of Space
____ 10. Find the area of a parallelogram that has the given vectors as adjacent sides. Round
your answer to two decimal places.
a.
b.
c.
d.
e.
____ 11. Find the area of the parallelogram with the vertices
, and . Round your answer to two decimal places.
a. 35.00
b. 3.00
c. 5.92
d. 7.00
e. 2.65
____ 12. Find the area of the triangle with the vertices and
. Round your answer to two decimal places.
a. 20.00
b. 8.00
c. 6.00
d. 2.24
e. 4.47
____ 13. A child applies the brakes on a bicycle by applying a downward force of 25 pounds
on the pedal when the crank makes a angle with the horizontal (see figure). The crank is 6 inches
in length. Find the torque at P. Round your answer to two decimal places.
11.4 The Cross Product of Two Vectors in Space 675
a. 12.50 ft-lb
b. 6.25 ft-lb
c. 75.00 ft-lb
d. 10.83 ft-lb
e. 21.65 ft-lb
____ 14. Both the magnitude and the direction of the force on a crankshaft change as the
crankshaft rotates. Find the torque on the crankshaft using the position and data shown in the figure.
Round your answer to two decimal places.
a. 2,598.08 ft-lb
b. 259.81 ft-lb
c. 212.13 ft-lb
d. 519.62 ft-lb
e. 150.00 ft-lb
____ 15. Find the triple scalar product .
a.
b.
c.
d.
e.
676 Chapter 11: Vectors and the Geometry of Space
____ 16. Use the triple scalar product to find the volume of the parallelepiped having adjacent
edges given by the vectors
a.
b.
c.
d. 516
e.
____ 17. Find the volume of the parallelepiped with the following vertices.
a.
b.
c.
d.
e.
11.4 The Cross Product of Two Vectors in Space 677
11.4 The Cross Product of Two Vectors in Space
Answer Section