Question

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

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

Short answer

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

Step-by-step solution

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. 

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