Q. 32 - Calculus

Question

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

$v\times w \mathrm{and} w\times v$

Step-by-Step Solution

Answer

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

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

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

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 

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

Step 5. Now solving the matrix.

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