Cross Products Lecture 19 Fri, Oct 21, 2005

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.

The Cross Product An easy way to remember the cross product. u1 u2 u3 v1 v2 v3

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

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

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

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.

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.

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

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

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

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