12.4: The Cross Product

University:
The Pennsylvania State University
Course:
Math 231 | Calculus Severable Variables
Academic year:

2024
Views:

441

Pages:

2
Author:

Alexis H.

12.4 the cross product Some applications of vectors require another operation called the cross product. In physics and engineering, the cross product is used to compute torque, a twisting force that causes an object to rotate Unlike the dot product (which is scalar), the cross product is a vector. It is defined algebraically using what is known as a 3x3 "determinant" An n x n matrix is an array consisting of n rows and n columns of numbers (or vectors) ¯ ¯a ¯ ¯c ¯ b ¯¯ = ad ¡ bc d¯ The formula expresses the 3x3 determinant in terms of 2x2 determinants called minors. The minors are obtained by crossing out the first row and one of the three columns of the 3x3 matrix Definition: The cross product ¯ ¯i ¯ v £ w = ¯¯ v1 ¯w 1 j v2 w2 ¯ k ¯¯ ¯¯ v 2 v3 ¯¯ = ¯¯w2 w 3¯ Theorem 1: Geometric description of the cross product i ¯ ¯ ¯ ¯ v1 v3 ¯ ¯ w3¯ j ¡ ¯w1 ¯ ¯ ¯ ¯ v1 v3 ¯ ¯ w3¯ j + ¯w1 ¯ v2 ¯¯ w2¯ k v x w is orthogonal to v and w kvk kwk sin µ v x w has length {v, w, v x w} forms a right-handed system One of the most striking properties of the cross product is that it is anti-commutative. Reversing the order changes the sign: w x v = -v x w Theorem 2: Basic properties of the cross product w x v = -v x w vxv=0 v x w = 0 if and only if either w = scalar times v for some scalar or v = 0 (¸v) £ w = v £ (¸w) = ¸ (v £ w) (u + v) x w = u x w + v x w u x (v + w) = u x v + u x w The cross product of any two of the standard basis vectors i, j, and k is equal to the third, possibly with a minus sign The cross product can be used to demonstrate a significant property of large-scale motion in the earth's atmosphere. Meteorologists study the properties and motion of small volumes of air to help forecast the weather and understand our climate Cross Products, Area, and Volume Scalar triple product u = hu1; u2; u3i v = hv1; v2; v3i w = hw1; w2; w3i µ¯ ¯ v2 u ¢ (v £ !) = u ¢ ¯¯w 2 ¯ ¯ ¯ ¯ v1 v3 ¯ ¯ w 3¯ i ¡ ¯ w 1 ¯ ¯ ¯ ¯ v1 v3 ¯ ¯ w 3¯ j + ¯ w 1 ¯ ¶ v2 ¯¯ w2¯ k