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.

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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.

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Question 1 In which direction will the cross product of k x i be oriented?

cannot be determined

-j

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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

i, the beam will not rotate

-j, the beam will not rotate

-k, the beam will rotate clockwise

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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

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Question 4 Determine the magnitude of the torque resulting from a force of F = <-28,0,0> applied at a radial displacement of r = <1,0,0> from a pivot point.

-28 mN

28 mN

0 mN

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Question 5 Determine the magnitude of the torque resulting from a force of F = <-28,24,0> applied at a radial displacement of r = <0,1,0> from a pivot point.

0 mN

52 mN

24 mN

28 mN

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Question 6 Determine the direction of the torque produced in Question 5.

-i

no rotation is produced

-k