Practice Problems
Basic Vector Cross Products
Directions:
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
.
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 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?
-j
j
cannot be determined
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.
-j
, the beam will not rotate
-k
, the beam will rotate clockwise
+k
, the beam will rotate counterclockwise
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.
-j
, the beam will not rotate
+k
, the beam will rotate counterclockwise
-k
, the beam will rotate clockwise
i
, the beam will not rotate
omit
Question 4
Determine the magnitude of the torque resulting from a force of
F = <-21,0,0>
applied at a radial displacement of
r = <1,0,0>
from a pivot point.
0 mN
-21 mN
21 mN
omit
Question 5
Determine the magnitude of the torque resulting from a force of
F = <-21,38,0>
applied at a radial displacement of
r = <0,1,0>
from a pivot point.
0 mN
38 mN
59 mN
21 mN
omit
Question 6
Determine the direction of the torque produced in Question 5.
-i
j
-k
+k
no rotation is produced
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Catharine H. Colwell
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Mark Acton