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Solve the following.
If a⃗ × b⃗ = a⃗ × c⃗ where a⃗, b⃗ and c⃗ are non-zero vectors, then prove that either b⃗ = c⃗ or a⃗…
If $\vec{a} \times \vec{b} = \vec{a} \times \vec{c}$ where $\vec{a}$, $\vec{b}$ and $\vec{c}$ are non-zero vectors, then prove that either $\vec{b} = \vec{c}$ or $\vec{a}$ and $(\vec{b} - \vec{c})$ are parallel.
Given: $\vec{a} \times \vec{b} = \vec{a} \times \vec{c}$
To show: $\vec{b} = \vec{c}$ or $\vec{a}$ and $(\vec{b} - \vec{c})$ are parallel
Answer
Answer
To prove: $\vec a \times \vec b = \vec a \times \vec c$ implies $\vec b = \vec c$ or $\vec a \parallel (\vec b - \vec c)$.
- $\vec a \times \vec b = \vec a \times \vec c$, given.
- $\vec a \times \vec b - \vec a \times \vec c = \vec 0$, transposing.
- $\vec a \times (\vec b - \vec c) = \vec 0$, by the distributive law of the cross product.
- $|\vec a||\vec b - \vec c|\sin\theta = 0$, taking magnitudes, where $\theta$ is the angle between $\vec a$ and $\vec b - \vec c$; $|\vec a| \neq 0$ as $\vec a$ is non-zero.
- So either $|\vec b - \vec c| = 0$, i.e. $\vec b = \vec c$, or $\sin\theta = 0$, i.e. $\theta = 0$ or $\pi$.
- $\sin\theta = 0$ means $\vec a$ and $\vec b - \vec c$ are parallel.
- Hence either $\vec b = \vec c$ or $\vec a$ and $(\vec b - \vec c)$ are parallel.