Week 14, Day 2

Today we learned about the cross product of two vectors. Let u = u1i + u2j + u3k and 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 and v. 

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 having u 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, and w as adjacent edges. 

Geometric Property of Triple Scalar Product:

Example 4: Volume by the Triple Scalar Product

Find the volume of the parallelpiped having

u = 3i - 5j + k, v = 2j - 2k, and w = 3i + j + k as adjacent edges.