Q. 31 - Calculus

Question

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

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

Step-by-Step Solution

Answer

The value of  $u\times w=25i-35j-10k \mathrm{and} w\times u=-25i+35j+10k$

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 $u\times w \mathrm{and} w\times u$

Step 2. Firstly finding the value of u × w

The value of vectors $u=(-3,1,-4) \mathrm{and} w=(1,3,13)$

The cross product of 

$u\times w=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ -3& 1& -4\\ 1& 3& 13\end{array}\right]$

Step 3. Now solving the matrix.

$$\begin{gathered} u\times w=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ -3& 1& -4\\ 1& 3& 13\end{array}\right] \\ u\times w=\left(\right(1\left)\right(13)-(-4\left)\right(3\left)\right)i+\left(\right(-3\left)\right(13)-(-4\left)\right(1\left)\right)j+\left(\right(-3\left)\right(3)-(1\left)\right(1\left)\right)k \\ u\times w=(13+12)i+(-39+4)j+(-9-1)k \\ u\times w=25i-35j-10k \end{gathered}$$

Step 4. Now finding the value of w × u

The value of vectors $u=(-3,1,-4) \mathrm{and} w=(1,3,13)$

The cross product of

$w\times u=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 1& 3& 13\\ -3& 1& -4\end{array}\right]$

Step 5. Now solving the matrix.

$$\begin{gathered} w\times u=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 1& 3& 13\\ -3& 1& -4\end{array}\right] \\ w\times u=\left(\right(-4\left)\right(3)-(1\left)\right(13\left)\right)i+\left(\right(-4\left)\right(1)-(-3\left)\right(13\left)\right)j+\left(\right(1\left)\right(1)-(-3\left)\right(3\left)\right)k \\ w\times u=(-12-13)i+(-4+39)j+(1+9)k \\ w\times u=-25i+35j+10k \end{gathered}$$