Q. 17 - Calculus

Question

If u, v and w are three vectors in ${\mathrm{ℝ}}^3$, what is wrong with the expression $u\times v\times w$?

Step-by-Step Solution

Answer

The cross product is not associative, so we must specify whether we mean $(u\times v)\times w \mathrm{or} u\times (v\times w)$.

Step 1. Given Information

If u, v and w are three vectors in ${\mathrm{ℝ}}^3$, we have to explain what is wrong with the expression $u\times v\times w$.

Step 2. Let u , v and w be vectors in ℝ 3 . Then the expression is

$u·(v\times w)=v·(w\times u)=w·(u\times v)$

But the given expression is $u\times v\times w$ which is wrong expression because the cross product is not associative, so we must specify whether we mean $(u\times v)\times w \mathrm{or} u\times (v\times w)$