Question

Let \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) be three vectors such that \(\mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c}\) and \(\mathbf{a} \times \mathbf{b} \neq 0. Show \;that \;\mathbf{b} = \mathbf{c}\).

Hint:

Quick Tip: When vectors are equal to each other, their cross products with a common vector must also be equal. A non-zero cross product indicates that the vectors are not parallel to each other.

Answer 1

We are given that: \[ \mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c} \quad \text{and} \quad \mathbf{a} \times \mathbf{b} \neq 0 \] We subtract the two vector equations: \[ \mathbf{a} \times \mathbf{b} - \mathbf{a} \times \mathbf{c} = 0 \] Using the distributive property of the cross product: \[ \mathbf{a} \times (\mathbf{b} - \mathbf{c}) = 0 \] For the cross product to be zero, the vectors \( \mathbf{a} \) and \( \mathbf{b} - \mathbf{c} \) must be parallel. This implies that: \[ \mathbf{b} - \mathbf{c} = \lambda \mathbf{a} \quad \text{for some scalar} \ \lambda \] Thus, we have: \[ \mathbf{b} = \mathbf{c} + \lambda \mathbf{a} \] Now, since \( \mathbf{a} \times \mathbf{b} \neq 0 \), we know that \( \mathbf{a} \) is not perpendicular to \( \mathbf{b} \), so \( \mathbf{b} \neq \mathbf{c} \). Therefore, we have shown that: \[ \mathbf{b} = \mathbf{c} \]