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The Vector Cross Product

Definition

The cross product of two 3-d vectors becomes a

3-d vector itself. The cross product is

Example

Method 1

An easier method is to use the formula listed in

the matrices part of the formula sheet.

Method 2

Applications of vxw - the perpendicular vector

The 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

Proof

Expand and simplify

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.

The area is given as follows

and

Now find the cross product of the two vectors

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).