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12.4 The Cross Product

12.4 The Cross Product - Math 224 Calculus III 12.4 The...

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Math 224 Calculus III 1 12.4 The Cross Product Recommended Homework: #1-19, 29-41 The cross product of 1 2 3 , , a a a a = and 1 2 3 , , b b b b = is defined to be: 2 3 3 2 3 1 1 3 1 2 2 1 , , a b a b a b ab ab a b a b = OR 2 3 3 2 3 1 1 3 1 2 2 1 ( ) ) ( ) a b a b a b a b a b a b a b = i +( j k Notes : This definition is not particularly memorable computationally; we will have a better way to remember it! The cross product is only defined for 3 The result of a cross product is a vector (unlike the result of a dot product). To motivate the idea, we consider the concept of torque from physics: Example: When we turn a bolt by applying a force (vector!) F to the end of a wrench, the result is to drive the bolt forward (or backwards) along its axis. The vector that describes this result is torque and the torque is found as the cross product of the length vector of the wrench and F . Finding the Cross Product The cross product is more easily remembered as the determinant of a 3 x 3 matrix. Determinant of a 2 x 2 matrix : The determinant of a b c d = a b c d = Determinant of a 3 x 3 matrix :
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