1. Cross Products
Lecture 19
Fri, Oct 21, 2005

2. u v = (u2v3 – u3v2, u3v1 – u1v3, u1v2 – u2v1).
The Cross Product
Let u = (u1, u2, u3) and v = (v1, v2, v3).
The cross product of u and v is defined to be
u v = (u2v3 – u3v2, u3v1 – u1v3, u1v2 – u2v1).
Note that the cross product is a vector, not a scalar.

3. The Cross Product An easy way to remember the cross product. u1 u2 u3v1 v2 v3

4. The Cross Product An easy way to remember the cross product. u1 u2 u3  u1 u2 u3 v1 v2 v3
u2v3 – u3v2

5. The Cross Product An easy way to remember the cross product. u1 u2 u3  u1 u2 u3 v1 v2 v3
u2v3 – u3v2
u3v1 – u1v3

6. The Cross Product An easy way to remember the cross product. u1 u2 u3  u1 u2 u3 v1 v2 v3
u2v3 – u3v2
u3v1 – u1v3
u1v2 – u2v1

7. Algebraic Properties of the Cross Product
Let u, v, and w be vectors and let c be a real number and let  be the angle between u and v.

8. The Right-hand Rule
The right-hand rules helps us remember which way u  v points.
Arrange the thumb, index finger, and middle finger so that they are mutually orthogonal.
Let the thumb represent u and the index finger represent v.
Then the middle finger represents u  v.

9. Finding Surface Normals
Given a triangle ABC, find a unit normal to the surface. A B C

10. Finding Surface Normals
Form the vectors u = B – A and v = C – A. C v A u B

11. Finding Surface Normals
Find w = u  v. w C v A u B

12. Example Let A = (1, 1, 2), B = (3, 1, 5), and C = (1, 0, 4).
Then u = B – A = (2, 0, 3) and v = C – A = (0, –1, 2).
So w = u  v = (3, -4, -2).
The surface normal is