Q. 20 - Calculus

Question

If u, v, and w are three mutually orthogonal vectors in ${\mathrm{ℝ}}^3$, explain why $u\times (v\times w)=0$.

Step-by-Step Solution

Answer

$u\times (v\times w)=0$ because $u\times (v\times w)=(u·w)v-(u·v)w$ and u, v, and w are three mutually orthogonal vectors.

Step 1. Given Information

If u, v, and w are three mutually orthogonal vectors in ${\mathrm{ℝ}}^3$, explain why $u\times (v\times w)=0$.

Step 2. It is given that u , v , and w are three mutually orthogonal vectors in ℝ 3 .

As we know that $u\times (v\times w)=(u·w)v-(u·v)w$

Since u, v, and w are three mutually orthogonal so the $u·w \mathrm{and} u·v$ is zero.

So 

$$\begin{gathered} u\times (v\times w)=0\times v-0\times w \\ u\times (v\times w)=0 \end{gathered}$$