Equilibrium Of Forces - Vector Condition

Question

A body is in equilibrium under 3 forces A,B,C.Show that A x B = B x C = C x A (x represents cross product). Well I know the longish method of writing A= a i + b j + c k form and then balancing along x,y,z axes. But is there a shorter way?

Answer 1

If three vectors are in equilibrium, then there is one very important relationship between these vectors: that $\mathbf{A} + \mathbf{B} + \mathbf{C} = 0$.

Hence, $\mathbf{A} \times (\mathbf{A} + \mathbf{B} + \mathbf{C}) = \mathbf{A} \times \mathbf{A} + \mathbf{A} \times \mathbf{B} + \mathbf{A} \times \mathbf{C} = \mathbf{0} $

This means, $\mathbf{A} \times \mathbf{B} = -\mathbf{A} \times \mathbf{C}$

Or, $\mathbf{A} \times \mathbf{B} = \mathbf{C} \times \mathbf{A}$

You can complete the proof by doing $\mathbf{B} \times (\mathbf{A} + \mathbf{B} + \mathbf{C}) = \mathbf{0} $