Chapter 9.7, Problem 60AYU

Prove property (3).

To determine

To prove: For vectors $u$ and $v$ that

$\text{u} \times \text{v} = - \left(\text{v} \times \text{u}\right)$

Answer to Problem 60AYU

$\text{u} \times \text{v} = - \left(\text{v} \times \text{u}\right)$

Explanation of Solution

Proof:

Let $\text{u} = {\text{a}}_1\text{i} + {\text{b}}_1\text{j} + {\text{c}}_1\text{k}$ and $\text{v} = {\text{a}}_2\text{i} + {\text{b}}_2\text{j} + {\text{c}}_2\text{k}$, then

$\begin{array}{lll}\text{u} \times \text{v}\hfill & = \hfill & \left|\begin{array}{ccc}\text{i}& \text{j}& \text{k}\\ {\text{a}}_1& {\text{b}}_1& {\text{c}}_1\\ {\text{a}}_2& {\text{b}}_2& {\text{c}}_2\end{array}\right|\hfill \\ \hfill & = \hfill & \left({\text{b}}_1{\text{c}}_2 - {\text{b}}_2{\text{c}}_1\right)\text{i} - \left({\text{a}}_1{\text{c}}_2 - {\text{a}}_2{\text{c}}_1\right)\text{j} + \left({\text{a}}_1{\text{b}}_2 - {\text{a}}_2{\text{b}}_1\right)\text{k}\hfill \end{array}$

$\begin{array}{lll}\text{v} \times \text{u}\hfill & = \hfill & \left|\begin{array}{ccc}\text{i}& \text{j}& \text{k}\\ {\text{a}}_2& {\text{b}}_2& {\text{c}}_2\\ {\text{a}}_1& {\text{b}}_1& {\text{c}}_1\end{array}\right|\hfill \end{array}$

$\begin{array}{ll} = \hfill & \left({\text{b}}_2{\text{c}}_1 - {\text{b}}_1{\text{c}}_2\right)\text{i} - \left({\text{a}}_2{\text{c}}_1 - {\text{a}}_1{\text{c}}_2\right)\text{j} + \left({\text{a}}_2{\text{b}}_2 - {\text{a}}_1{\text{b}}_2\right)\text{k}\hfill \end{array}$

$\begin{array}{ll} = \hfill & - \left[\left({\text{b}}_1{\text{c}}_2 - {\text{b}}_2{\text{c}}_1\right)\text{i} - \left({\text{a}}_1{\text{c}}_2 - {\text{a}}_2{\text{c}}_1\right)\text{j} + \left({\text{a}}_1{\text{b}}_2 - {\text{a}}_2{\text{b}}_1\right)\text{k}\right]\hfill \end{array}$

Which is equal to $- \left(\text{u} \times \text{v}\right)$

Hence $\text{u} \times \text{v} = - \left(\text{v} \times \text{u}\right)$