1. Discrete Math Section 12.9 Define and apply the cross product The cross product of two vectors results in another vector. If v 1 = and v 2 =, the cross product v 1 x v 2 is found by: | i j k | | a 1 b 1 c 1 | | a 2 b 2 c 2 | Example: If v 1 = and v 2 =, find v 1 x v 2

2. Properties of cross products 1. u x v is perpendicular to u and v. 2. v x u = -(u x v) or u x v and v x u have opposite directions. 3. sin Θ = |u x v | where Θ is the angle |u| |v | between u and v. 4. u x (v + w) = (u x v) + (u x w) 5. u is parallel to v iff u x v = 0 6. | u x v | is the area of the parallelogram formed by u and v.

3. example A. Find a vector perpendicular to the plane containing the points A(5,6,2), B(4,-1,0), and C(2,5,1). B. Find the area of the parallelogram with sides formed by and C. Find the angle between and

4. Assignment Page 467 Problems 2, 4, 6, 10a,c, 12a,c, 13a