Consider the vector products and Give an example that illustrates the general rule that these two vector products do not have the same magnitude or direction. Can you choose vectors such that these two vector products are equal? If so, give an example.
Solution 23DQ Part A: Step 1: Any three random vector we will take say A,B and C . Among these let’s take B and C lie on one plane and the vector Alie on a plane perpendicular to the plane where B and C reside. The diagram is as follows, The vector A points into the page, perpendicular to the plane where B and C vector lie. Step 2: First let’s find A × (B × C). (B × C) gives a new vector Dsay, in the direction which is along the vector A by the right hand screw rule of vector cross product. As, vector A and vector D are in same direction, the angle between them will be 0 . So, , which is a null vector. Step 3: Now let’s calculate (A × B) × C . Here A × B will give a new vector E say. This vector will lie on the same plane where vector B and vector C lie. The magnitude will be, 0 E = A × B = A| || | 90 = A B ×| || | B . | || | Because the angle between vector A and vector B is 90 and sin 90 =1 we know. Now (A × B) × C = E × C = E C | || |= A B C | || || | Where is the angle between vector E and vector C. It can be clearly seen that, (A × B) × C / 0 . So, we proved that A × (B × C) = / (A × B) × C . Part B: Step 1: The above equation is true for any randomly chosen vectors. But for a very special case the magnitude and direction are same for both the sides. Let’s show it. Let’s take 3 mutually perpendicular vectors A,B and C along the 3 axes of a cartesian coordinate system respectively. 0 0 So, A × (B × C) = A × ( | || |n 90 ) = A × ( B| || |as sin 90 = 1. 0 A × (B × C) = A | || || | = 0. This is because, the vector formed by B × C is along the vector A and the angle between 0 them will be zero. And we know that sin 0 = 0. So, A × (B × C) = 0