Q. 75 - Calculus

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 $v=w$.

Step-by-Step Solution

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 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 ℝ 3 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 $v-w=0$.