## Problem on whether two vectors can have the same cross product

If $\bfi \times \bfx = \bfi \times \bfy$, does $\bfx = \bfy$? If so, prove it. If not, find a counterexample.
• ## Solution

We are trying to determine whether two vectors can have the same cross product with $\bfi$.
For problems where we need decide if a statement is true, we should start by exploring the problem in simple cases.
We let $\bfy$ be the simplest of all vectors: $\bfy = \mathbf{0}$.
Because $\bfi \times \mathbf{0} = \mathbf{0}$, the problem becomes:
If $\bfi \times \bfx=\mathbf{0}$, does $\bfx = \mathbf{0}$?
Recall that
Hence, if $\bfx$ is a multiple of $\bfi$ then $\bfi \times \bfx = \mathbf{0}.$
We have thus found a counterexample: $\bfx = \bfi$ and $\bfy = \mathbf{0}$.
The statement is not true because $\bfi \times \bfi = \bfi \times \mathbf{0}$ and $\bfi \neq \mathbf{0}$.