Chapter 11.4, Problem 45ES

Expressions of the form $\text{u}\times \left(\text{v}\times \text{w}\right)\text{ and }\left(\text{u}\times \text{v}\right)\times \text{w}$ are called vector triple products. It can be proved with some effort that

$\begin{array}{l}\text{u}\times \left(\text{v}\times \text{w}\right)=\left(\text{u}\cdot \text{w}\right)\text{v}-\left(\text{u}\cdot \text{v}\right)\text{w}\\ \left(\text{u}\times \text{v}\right)\times \text{w}=\left(\text{w}\cdot \text{u}\right)\text{v}-\left(\text{w}\cdot \text{v}\right)\text{u}\end{array}$

These expressions can be summarized with the following mnemonic rule: $\text{vector triple product = }\left(\text{outer}\cdot \text{remote}\right)\text{adjacent}-\left(\text{outer adjacent}\right)\text{remote}$ See if you can figure out what the expressions "outer," "remoter and "adjacent" mean in this rule, and then use the rule to find the two vector triple products of the vectors $\text{u}=\text{i}+\text{3j}-\text{k,}\text{v}=\text{i}+\text{j}+\text{2k,}\text{w}=\text{3i}-\text{j}+\text{2k}$