Q. 23 - Calculus

Question

In Exercises 22–29 compute the indicated quantities when $u=(2,1,-3), v=(4,0,1), \mathrm{and} w=(-2,6,5)$

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

Step-by-Step Solution

Answer

The value of $u\times w=23i+4j+14k$ and $w\times u=-23i-4j-14k$.

Step 1. Given Information

In Exercises 22–29 compute the indicated quantities when$u=(2,1,-3), v=(4,0,1), \mathrm{and} w=(-2,6,5)$

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 $u=(2,1,-3), w=(-2,6,5)$

The cross product of $u\times w=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 2& 1& -3\\ -2& 6& 5\end{array}\right]$

Step 3. Now solving the matrix.

$$\begin{gathered} u\times w=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 2& 1& -3\\ -2& 6& 5\end{array}\right] \\ u\times w=\left(\right(1\left)\right(5)-(-3\left)\right(6\left)\right)i+\left(\right(2\left)\right(5)-(-2\left)\right(-3\left)\right)j+\left(\right(2\left)\right(6)-(1\left)\right(-2\left)\right)k \\ u\times w=(5+18)i+(10-6)j+(12+2)k \\ u\times w=23i+4j+14k \end{gathered}$$

Step 4. Firstly finding the value of w × u

The value of vectors $w=(-2,6,5) \mathrm{and} \mathrm{u}=(2,1,-3)$

The cross product of $w\times u=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ -2& 6& 5\\ 2& 1& -3\end{array}\right]$

Step 5. Now solving the matrix.

$$\begin{gathered} w\times u=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ -2& 6& 5\\ 2& 1& -3\end{array}\right] \\ w\times u=\left(\right(6\left)\right(-3)-(5\left)\right(1\left)\right)i+\left(\right(-2\left)\right(-3)-(5\left)\right(2\left)\right)j+\left(\right(-2\left)\right(1)-(6\left)\right(2\left)\right)k \\ w\times u=(-18-5)i+(6-10)j+(-2-12)k \\ w\times u=-23i-4j-14k \end{gathered}$$