Chapter 12, Problem 17AAE

(a)

To determine

To prove: The formula $\left(\overrightarrow{u}\times \overrightarrow{v}\right)\times \overrightarrow{w}=\left(\overrightarrow{u}\cdot \overrightarrow{w}\right)\overrightarrow{v}-\left(\overrightarrow{v}\cdot \overrightarrow{w}\right)\overrightarrow{u}$ and $\overrightarrow{u}\times \left(\overrightarrow{v}\times \overrightarrow{w}\right)=\left(\overrightarrow{u}\cdot \overrightarrow{w}\right)\overrightarrow{v}-\left(\overrightarrow{u}\cdot \overrightarrow{v}\right)\overrightarrow{w}$ for the vectors $\overrightarrow{u}=2\overrightarrow{i},\overrightarrow{v}=2\overrightarrow{j},\overrightarrow{w}=2\overrightarrow{k}$ by evaluating both the sides and comparing the results.

(b)

To determine

To prove: The formula $\left(\overrightarrow{u}\times \overrightarrow{v}\right)\times \overrightarrow{w}=\left(\overrightarrow{u}\cdot \overrightarrow{w}\right)\overrightarrow{v}-\left(\overrightarrow{v}\cdot \overrightarrow{w}\right)\overrightarrow{u}$ and $\overrightarrow{u}\times \left(\overrightarrow{v}\times \overrightarrow{w}\right)=\left(\overrightarrow{u}\cdot \overrightarrow{w}\right)\overrightarrow{v}-\left(\overrightarrow{u}\cdot \overrightarrow{v}\right)\overrightarrow{w}$ for the vectors $\overrightarrow{u}=\overrightarrow{i}-\overrightarrow{j}+\overrightarrow{k},\overrightarrow{v}=2\overrightarrow{i}+\overrightarrow{j}-2\overrightarrow{k},\overrightarrow{w}=-\overrightarrow{i}+2\overrightarrow{j}-\overrightarrow{k}$ by evaluating both the sides and comparing the results.

(c)

To determine

To prove: The formula $\left(\overrightarrow{u}\times \overrightarrow{v}\right)\times \overrightarrow{w}=\left(\overrightarrow{u}\cdot \overrightarrow{w}\right)\overrightarrow{v}-\left(\overrightarrow{v}\cdot \overrightarrow{w}\right)\overrightarrow{u}$ and $\overrightarrow{u}\times \left(\overrightarrow{v}\times \overrightarrow{w}\right)=\left(\overrightarrow{u}\cdot \overrightarrow{w}\right)\overrightarrow{v}-\left(\overrightarrow{u}\cdot \overrightarrow{v}\right)\overrightarrow{w}$ for the vectors $\overrightarrow{u}=2\overrightarrow{i}+\overrightarrow{j},\overrightarrow{v}=2\overrightarrow{i}-\overrightarrow{j}+\overrightarrow{k},\overrightarrow{w}=\overrightarrow{i}+2\overrightarrow{k}$ by evaluating both the sides and comparing the results.

(d)

To determine

To prove: The formula $\left(\overrightarrow{u}\times \overrightarrow{v}\right)\times \overrightarrow{w}=\left(\overrightarrow{u}\cdot \overrightarrow{w}\right)\overrightarrow{v}-\left(\overrightarrow{v}\cdot \overrightarrow{w}\right)\overrightarrow{u}$ and $\overrightarrow{u}\times \left(\overrightarrow{v}\times \overrightarrow{w}\right)=\left(\overrightarrow{u}\cdot \overrightarrow{w}\right)\overrightarrow{v}-\left(\overrightarrow{u}\cdot \overrightarrow{v}\right)\overrightarrow{w}$ for the vectors $\overrightarrow{u}=\overrightarrow{i}+\overrightarrow{j}-2\overrightarrow{k},\overrightarrow{v}=-\overrightarrow{i}-\overrightarrow{k},\overrightarrow{w}=2\overrightarrow{i}+4\overrightarrow{j}-2\overrightarrow{k}$ by evaluating both the sides and comparing the results.