Chapter 9.7, Problem 20AYU

In Problems 15-22, find (a) $v\times w$, (b) $w\times v$, (c) $w\times w$, and (d) $v\times v$.

$\begin{array}{l}v\text{ = 3}i+j\text{ + 3}k\\ w\text{ = }i-k\end{array}$

To determine

To find: The value of

a. $\text{v} \times \text{w}$,

b. $\text{w} \times \text{v}$,

c. $\text{w} \times \text{w}$, and

d. $\text{v} \times \text{v}$.

For, $\text{v} = 3\text{i} + \text{j} + 3\text{k}$

$\text{w} = \text{i} - \text{k}$

Answer to Problem 20AYU

Solution:

a. $\text{v} \times \text{w} = - \text{i} + 6\text{j} - \text{k}$,

b. $\text{w} \times \text{v} = \text{i} - 6\text{j} + \text{k}$,

c. $\text{w} \times \text{w} = 0$, and

d. $\text{v} \times \text{v} = 0$

Explanation of Solution

Given:

$\begin{array}{lll}\text{v}\hfill & = \hfill & \text{3i} + \text{j} + 3\text{k}\hfill \\ \text{w}\hfill & = \hfill & \text{i} - \text{k}\hfill \end{array}$

Formula used:

The cross product of the vector $\text{v} = {\text{a}}_1\text{i} + {\text{b}}_1\text{j} + {\text{c}}_1\text{k}$, and $\text{w} = {\text{a}}_2\text{i} + {\text{b}}_2\text{j} + {\text{c}}_2\text{k}$ is

$\text{v} \times \text{w} = \left|\begin{array}{ccc}\text{i}& \text{j}& \text{k}\\ {\text{a}}_1& {\text{b}}_1& {\text{c}}_1\\ {\text{a}}_2& {\text{b}}_2& {\text{c}}_2\end{array}\right| = \left|\begin{array}{cc}{\text{b}}_1& {\text{c}}_1\\ {\text{b}}_2& {\text{c}}_2\end{array}\right|\text{i} - \left|\begin{array}{cc}{\text{a}}_1& {\text{c}}_1\\ {\text{a}}_2& {\text{c}}_2\end{array}\right|\text{j} + \left|\begin{array}{cc}{\text{a}}_1& {\text{b}}_1\\ {\text{a}}_2& {\text{b}}_2\end{array}\right|\text{k}$

Calculation:

a. $\begin{array}{lll}\text{v} \times \text{w}\hfill & = \hfill & \left|\begin{array}{ccc}\text{i}& \text{j}& \text{k}\\ 3& 1& 3\\ 1& 0& - 1\end{array}\right|\hfill \\ \hfill & = \hfill & \left|\begin{array}{cc}1& 3\\ 0& - 1\end{array}\right|\text{i} - \left|\begin{array}{cc}3& 3\\ 1& - 1\end{array}\right|\text{j} + \left|\begin{array}{cc}3& 1\\ 1& 0\end{array}\right|\text{k}\hfill \\ \hfill & = \hfill & - \text{i} + 6\text{j} - \text{k}\hfill \end{array}$

b. $\begin{array}{lll}\text{w} \times \text{v}\hfill & = \hfill & \left|\begin{array}{ccc}\text{i}& \text{j}& \text{k}\\ 1& 0& - 1\\ 3& 1& 3\end{array}\right|\hfill \\ \hfill & = \hfill & \left|\begin{array}{cc}0& - 1\\ 1& 3\end{array}\right|\text{i} - \left|\begin{array}{cc}1& - 1\\ 3& 3\end{array}\right|\text{j} + \left|\begin{array}{cc}1& 0\\ 3& 1\end{array}\right|\text{k}\hfill \\ \hfill & = \hfill & \text{i} - 6\text{j} + \text{k}\hfill \end{array}$

c. $\begin{array}{lll}\text{w} \times \text{w}\hfill & = \hfill & \left|\begin{array}{ccc}\text{i}& \text{j}& \text{k}\\ 1& 0& - 1\\ 1& 0& - 1\end{array}\right|\hfill \\ \hfill & = \hfill & \left|\begin{array}{cc}0& - 1\\ 0& - 1\end{array}\right|\text{i} - \left|\begin{array}{cc}1& - 1\\ 1& - 1\end{array}\right|\text{j} + \left|\begin{array}{cc}1& 0\\ 1& 0\end{array}\right|\text{k}\hfill \\ \hfill & = \hfill & 0\text{i} - 0\text{j} + 0\text{k} = 0\hfill \end{array}$

d. $\begin{array}{lll}\text{v} \times \text{v}\hfill & = \hfill & \left|\begin{array}{ccc}\text{i}& \text{j}& \text{k}\\ 3& 1& 3\\ 3& 1& 3\end{array}\right|\hfill \\ \hfill & = \hfill & \left|\begin{array}{cc}1& 3\\ 1& 3\end{array}\right|\text{i} - \left|\begin{array}{cc}3& 3\\ 3& 3\end{array}\right|\text{j} + \left|\begin{array}{cc}3& 1\\ 3& 1\end{array}\right|\text{k}\hfill \\ \hfill & = \hfill & 0\text{i} - 0\text{j} + 0\text{k} = 0\hfill \end{array}$