Question

Let u, v, andw be vectors in ${\mathrm{ℝ}}^3$. Prove that $u\times v=u\times w$if and only if u is parallel to $v-w$.

Short answer

Hence, we prove that$u\times v=u\times w$ if and only if u is parallel to $v-w$.

Step-by-step solution

Step 1. Given Information

Let u, v, and w be vectors in ${\mathrm{ℝ}}^3$. Prove that $u\times v=u\times w$ if and only if u is parallel to $v-w$.

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

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

That means$u\parallel v-w$.