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

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.

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

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