Question

Let u, v, and w be vectors in ${\mathrm{ℝ}}^3$with $u\ne 0$. Show that if $u\times v=u\times w$and $u·v=u·w$, then $v=w$.

Short answer

If$u\times v=u\times w$then$u\parallel v-w$.

If $u·v=u·w$ then u is orthogonal to $v-w$this can only happen if$v-w=0$

Step-by-step solution

Step 1. Given Information

Let u, v, and w be vectors in ${\mathrm{ℝ}}^3$with$u\ne 0$. Show that if $u\times v=u\times w$and $u·v=u·w$, then $v=w$.

Step 2. Let u, v, and w be vectors in ℝ3 with u≠0

$$\begin{gathered} u\times v=u\times w \\ \mathrm{Subtract}u\times w\mathrm{on}\mathrm{both}\mathrm{side} \\ \mathrm{u}\times \mathrm{v}-\mathrm{u}\times \mathrm{w}=\mathrm{u}\times \mathrm{w}-\mathrm{u}\times \mathrm{w} \\ \mathrm{u}\times \mathrm{v}-\mathrm{u}\times \mathrm{w}=0 \\ \mathrm{u}\times (\mathrm{v}-\mathrm{w})=0 \\ \mathrm{u}=0\mathrm{v}-\mathrm{w}=0 \end{gathered}$$

That means$u\parallel v-w$

Step 3. Now if u·v=u·w

If $u·v=u·w$, then u is orthogonal to $v-w$. Thus, $v-w$is orthogonal to itself. This can only happen if $v-w=0$.