Q. 30 - Calculus

Question

In Exercises 30–35 compute the indicated quantities when 

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

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

Step-by-Step Solution

Answer

The value of $u\times v=5i-7j-2k \mathrm{and} v\times u=-5i+7j+2k$.

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 $u\times v \mathrm{and} v\times u$.

Step 2. Firstly finding the value of u × v

The value of vectors $u=(-3,1,-4), v=(2,0,5)$

The cross product of $u\times v$

$u\times v=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ -3& 1& -4\\ 2& 0& 5\end{array}\right]$

Step 3. Now solving the matrix.

$$\begin{gathered} u\times v=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ -3& 1& -4\\ 2& 0& 5\end{array}\right] \\ u\times v=\left(\right(1\left)\right(5)-(-4\left)\right(0\left)\right)i+\left(\right(-3\left)\right(5)-(-4\left)\right(2\left)\right)j+\left(\right(-3\left)\right(0)-(1\left)\right(2\left)\right)k \\ u\times v=(5-0)i+(-15+8)j+(0-2)k \\ u\times v=5i-7j-2k \end{gathered}$$

Step 4. Now finding the value of v × u

The value of vectors $u=(-3,1,-4), v=(2,0,5)$

The cross product of 

$v\times u=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 2& 0& 5\\ -3& 1& -4\end{array}\right]$

Step 5. Now solving the matrix.

$$\begin{gathered} v\times u=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 2& 0& 5\\ -3& 1& -4\end{array}\right] \\ v\times u=\left(\right(0\left)\right(-4)-(5\left)\right(1\left)\right)i+\left(\right(2\left)\right(-4)-(-3\left)\right(5\left)\right)j+\left(\right(2\left)\right(1)-(-3\left)\right(0\left)\right)k \\ v\times u=(0-5)i+(-8+15)j+(2-0)k \\ v\times u=-5i+7j+2k \end{gathered}$$