Question

Find the value of: \[ \mathbf{a} \times (\mathbf{b} + \mathbf{c}) + \mathbf{b} \times (\mathbf{c} + \mathbf{a}) + \mathbf{c} \times (\mathbf{a} + \mathbf{b}) \]

Hint:

The identity \( \mathbf{x} \times \mathbf{y} = -(\mathbf{y} \times \mathbf{x}) \) helps in simplifying vector cross products.

Answer 1

Step 1: Expand the expression using distributive property of cross product. \[ \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c} + \mathbf{b} \times \mathbf{c} + \mathbf{b} \times \mathbf{a} + \mathbf{c} \times \mathbf{a} + \mathbf{c} \times \mathbf{b} \] Step 2: Use the identity \( \mathbf{x} \times \mathbf{y} = -(\mathbf{y} \times \mathbf{x}) \). Pairs of terms cancel: \[ \mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{a} = 0, \quad \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{b} = 0, \quad \mathbf{c} \times \mathbf{a} + \mathbf{a} \times \mathbf{c} = 0 \] Step 3: Conclude result. \[ \mathbf{a} \times (\mathbf{b} + \mathbf{c}) + \mathbf{b} \times (\mathbf{c} + \mathbf{a}) + \mathbf{c} \times (\mathbf{a} + \mathbf{b}) = 0 \]