Aidan's Math Blog: Cross Product of Two Vectors

Today we learned about one of the hardest things ever to cross a papers face. I can assure you this is quite a bit of information so hold ok to your seats folks. Also much of this is from the book so my apologies if you cannot understand the written out verbage. Hopefully the pictures help!

Let u = u1i + u2j + u3kand v = v1i + v2j + v3k be vectors in space. The cross product of u and v is the vector. u x v = (u2v3 - u3v2)i - (u1v3 - u3v1)j + (u1v2 - u2v1)k. A convenient way to calculate u x v is to use the following determinant form with cofactor expansion. (This 3 x 3 determinant form is used simply to help remember the formula for the cross product--it is technically not a determinant because not all the entries of the corresponding matrix are real numbers).

Example 1: Finding Cross Products

Given u = i + 2j + k and v = 3i + j + 2k, find the following:

a) u x v

b) v x u

c) v x v

Algebraic Properties of the Cross Product

1) u x v = -(v x u)

2) u x (v + w) = (u x v) + (u x w)

3) c (u x v) = (cu) x v = u x (cv)

4) u x 0 = 0 x u = 0

5) u x u = 0

6) u x (v x w) = (u x v) x w

Geometric Properties of the Cross Product

This property indicates that the vectors u x v and v x u have equal lengths but opposite directions.

Let u and v be nonzero vectors in space, and let theta be the angle between u andv.

1) u x v is orthogonal to both u and v.

2) ||u x v|| = ||u|| ||v|| sin theta.

3) u x v = 0 if and only if u and v are scalar multiples.

4) ||u x v|| = area of parallelogram havingu and v as adjacent sides.

Example 2: Using the Cross Product

Find a unit vector that is orthogonal to both

u = 3i - 4j + k and v = -3i + 6j

The Triple Scalar Product

For vectors u, v, and w in space, the dot product of u and v x w

is called the triple scalar product of u, v,and w.

If the vectors u, v, and w do not lie in the same plane, the triple scalar product u x (v x w) can be used to determine the volume of the parallelpiped with u, v, andw as adjacent edges.