The Vector Cross Product

1
The Vector Cross Product
2
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
3
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
4
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
5
So that proof means
The area of a triangle where a and b are vectors
is given by
6
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
7
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).