The Vector Cross Product

DefinitionThe cross product of two 3-d vectors becomes a 3-d vector itself.The cross product is:

An easier method is to use the formula listed in the matrices part of the formula sheet.

Example

Method 1

Method 2

Applications of vxw - the perpendicular vectorThe resulting vector of the cross product of vxw will always be

perpendicular to v and w, as shown in the diagram below.Find the vector that is perpendicular to:

1. and

2. 3i + 2j - k and 3j - 5k.

-7i + 15j + 9k

Applications of vxw - the area of a triangle or parallelogram

Area of a triangle is given by . If a and b are vectors the area of a triangle becomes:

Expand and simplify

Proof:

So that proof means:

The area of a triangle where a and b are vectors is given by:

Find the area of the triangle with vertices A(1,1,2) B(-1,3,2) and C(4,1,5).

Find two sides of the triangle, the vectors AB and AC.

and

Now find the cross product of the two vectors:

The area is given as follows:

TRY THESE YOURSELF

Find the area of the triangle with vertices A(2,7,3), B(-8,3,1) and C(1,6,-2).

Find two perpendicular unit vectors to the plane containing the points (4,1,3), (1,1,2) and (7,2,-4).