Question

In Exercises 30–35 compute the indicated quantities when $u=(-3,1,-4),v=(2,0,5),\mathrm{and}w=(1,3,13)$

role="math" localid="1649434688557" $v\times w\mathrm{and}w\times v$

Short answer

The value of$v\times w=-15i+21j+6k\mathrm{and}w\times v=15i-21j-6k$

Step-by-step solution

Step 1. Given Information 

The indicated quantities when $u=(-3,1,-4),v=(2,0,5),\mathrm{and}w=(1,3,13)$

We have to find the value of$v\times w\mathrm{and}w\times v$

Step 2. Firstly finding the value of v×w

The value of vectors$v=(2,0,5),\mathrm{and}w=(1,3,13)$

The cross product of

role="math" localid="1649434703174" $v\times w=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 2& 0& 5\\ 1& 3& 13\end{array}\right]$

Step 3. Now solving the matrix.

$$\begin{gathered} v\times w=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 2& 0& 5\\ 1& 3& 13\end{array}\right] \\ v\times w=\left(\right(0\left)\right(13)-(5\left)\right(3\left)\right)i+\left(\right(2\left)\right(13)-(5\left)\right(1\left)\right)j+\left(\right(2\left)\right(3)-(0\left)\right(1\left)\right)k \\ v\times w=(0-15)i+(26-5)j+(6-0)k \\ v\times w=-15i+21j+6k \end{gathered}$$

Step 4. Now finding the value of w×v

The value of vectors$v=(2,0,5),\mathrm{and}w=(1,3,13)$

The cross product of

role="math" localid="1649434718022" $w\times v=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 1& 3& 13\\ 2& 0& 5\end{array}\right]$

Step 5. Now solving the matrix.

$$\begin{gathered} w\times v=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 1& 3& 13\\ 2& 0& 5\end{array}\right] \\ w\times v=\left(\right(3\left)\right(5)-(13\left)\right(0\left)\right)i+\left(\right(1\left)\right(5)-(13\left)\right(2\left)\right)j+\left(\right(1\left)\right(0)-(3\left)\right(2\left)\right)k \\ w\times v=(15-0)i+(5-26)j+(0-6)k \\ w\times v=15i-21j-6k \end{gathered}$$