Problem on whether two vectors can have the same cross product

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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
    Two vectors have zero cross product if they are multiples of each other.
    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 $\bfi \neq \mathbf{0}$.

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