Cross Product

The cross product is vector multiplication used when the resulting product should also be a vector, and has a largest effect when the products are perpindicular. The cross product of vector A with vector B is spoken as "A cross B". The product has a magnitude of

\begin{displaymath}\bf A \times\bf B \rm =AB sin \theta\end{displaymath}

and has a direction which is at right angles to both vectors, and can be found using a "right hand rule", holding your hand flat with the thumb stuck out, pointing your fingers in the direction of the first vector with your palm facing the direction of the second vector. The direction of your thumb is the direction of the resultant vector.

The following graphic show a three dimensional perspective of the cross product. Vector A is along the positive y axis, and the image rotates about the y axis. The image is rotating clockwise about the y-axis, the right side of the image is moving away from you, and the left side is moving towards you.

three d image of cross product.

Physically, you can think of the magnitude of the cross product as being the area of a parallelogram made from the two vectors.

image shows two vectors, one along y axis and one in second quadrant, as two sides of a parrallelogram.

In coordinate notation, in three dimensions, this can also be written as

\begin{displaymath}\bf A x B \rm = [ A_y B_z - A_z B_y , A_z B_x - A_x B_z , A_x B_y - A_y B_x ] .\end{displaymath}

As an example, consider the torque, or twisting force, on a bolt by a wrench. The longer of a wrench that you use, the easier it is to loosen a bolt. The torque is generally written as

\begin{displaymath}\bf\tau = R \times F \rm .\end{displaymath}

The direction of R is along the wrench outward from the bolt to the handle. The direction of the torque can be found using a "right hand rule", where you point your flattened right hand with your fingers pointing in the direction of R, and your palm facing in the direction of F. Stick your thumb out without otherwise moving your hand, and the diection of your thumb will give you the direction of the torque. In this case, the direction of the torque gives you the axis about which the bolt will twist counter-clockwise.

Quick Quiz: You are trying to loosen a bolt, but cannot get a good grip due to a pipe that is in the way. You apply a force of 2 N, at an angle such that the angle in between the arm of the wrench and a line perpindicular to the wrench is 30 degrees. What is the torque applied to the bolt?
0.25 N.m
0.5 N.m
0 N.m
0.4 N.m

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