Practice Problems
Basic Vector Cross Products
Topics:
On this worksheet students will practice with the properties of
vector cross products
. This mathematical form is used when calculating
torques
,
anguluar momentum
, and
magnetic forces
.
Page Directions
The numerical values in this worksheet are randomly generated allowing students the opportunity to conveniently practice, and drill, common situations.
Before beginning any given worksheet, please look over all of the questions and make sure that there are
no duplicate
answers shown for the same question. If duplicates are present simply refresh the page until every answer is unique.
In order to check an answer
(even when you are just starting the worksheet on Question 1)
it is necessary to
any questions that you have not answered. Once you start submitting answers, the page may be checked as many times as necesasary without changing the randomized answers. Relevant scoring will be provided at the top of the page only when you answer all of the questions on your original submission.
Background:
The unit vector along the x-axis is
i = <1,0,0>
, along the y-axis is
j = <0,1,0>
, and along the z-axis is
k = <0,0,1>
. The right-hand rule for determining the direction of the cross product
C = A x B
, begins by placing the vectors concurrently, so that their tails start at the same position, then, with the fingers of your right hand pointing in the direction of vector
A
, curl your fingers towards vector
B
. Your thumb will naturally twist to point in the direction of vector
C
. The
applet on this page
will allow you to visualize the direction of
C = A x B
. For example:
i x j = k
j x i = - k
The magnitude of a vector cross product can also be calculated using
determinants
or the equation
where θ is the angle between vectors
r
and
F
(rotating from
r
towards
F
). The unit used to measure torque is a meter-newton (mN) which does
not
equal a Joule.
omit
Question 1
In which direction will the cross product of
k x i
be oriented?
cannot be determined
j
-j
omit
Question 2
A commonly used vector cross product in physics is for torque.
τ = r x F
. Determine the direction of the torque on a uniform horizontal beam pivoted at its center by a downward force applied at the right end of the beam.
The radius vector points from the pivot point to the application point of the force.
+k
, the beam will rotate counterclockwise
-k
, the beam will rotate clockwise
-j
, the beam will not rotate
i
, the beam will not rotate
omit
Question 3
Determine the direction of the torque on a uniform horizontal beam pivoted at its center by a downward force applied at the left end of the beam.
i
, the beam will not rotate
+k
, the beam will rotate counterclockwise
-k
, the beam will rotate clockwise
-j
, the beam will not rotate
omit
Question 4
Determine the magnitude of the torque resulting from a force of
F = <-25,0,0>
applied at a radial displacement of
r = <11,0,0>
from a pivot point.
275 mN
-275 mN
0 mN
omit
Question 5
Determine the magnitude of the torque resulting from a force of
F = <-25,22,0>
applied at a radial displacement of
r = <0,11,0>
from a pivot point.
0 mN
242 mN
517 mN
275 mN
omit
Question 6
Determine the direction of the torque produced in Question 5.
no rotation is produced
-i
-k
j
+k
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