 # PROOF THAT THE MAGNITUDE OF THE CROSS PRODUCT IS THE AREA OF THE PARALLELOGRAM

12 years ago

Dear student,

We define the cross product of two three-dimensional vectors a and b by the requirements:

• a × b is a vector that is perpendicular to both a and b.
• ||a × b|| is the area of the parallelogram spanned by a and b (i.e. the parallelogram whose adjacent sides are the vectors a and b).
• The direction of a×b is determined by the right-hand rule. (This means that if we curl the fingers of the right hand from a to b, then the thumb points in the direction of a × b.)

From simple trigonometry, we can calculate that the area of the parallelogram spanned by a and b is

 ||a|| ||b|| sin θ,

where θ is the angle between a and b. (One can view the parallelogram as having a base of length ||b|| and perpendicular height ||a|| sin θ, as pictured below.)

This formula shows that the magnitude of the cross product is largest when a and b are perpendicular. On the other hand, if a and b are parallel or if either vector is the zero vector, then the cross product is the zero vector. (It is a good thing that we get the zero vector so that the above definition still makes sense. If the vectors are parallel or one vector is the zero vector, then there is not a unique line perpendicular to both a and b. But since there is only one vector of zero length, the definition still uniquely determines the cross product.)

All the best.

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