Question

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

localid="1649405204459" role="math" $(u\times v)·w\mathrm{and}u·(v\times w)$

Short answer

The value of$(u\times v)·w=(-1,20,1)\mathrm{and}u·(v\times w)=(-4,23,21)$

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$(u\times v)·w\mathrm{and}u·(v\times w)$

Step 2. Firstly finding the value of (u×v)·w

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

Now solving the cross product.

localid="1649405220352" $$\begin{gathered} u\times v=\left(\right(1\left)\right(1)-(-3\left)\right(0\left)\right)i+\left(\right(2\left)\right(1)-(-3\left)\right(4\left)\right)j+\left(\right(2\left)\right(0)-(1\left)\right(4\left)\right)k \\ u\times v=(1-0)i+(2+12)j+(0-4)k \\ u\times v=1i+14j-4k \end{gathered}$$

Step 3. Now finding the value of (u×v)·w

Now the vectors of $u\times v\mathrm{is}(1,14,-4)$

role="math" localid="1649405226967" $$\begin{gathered} (u\times v)·w=(1,14,-4)·(-2,6,5) \\ (u\times v)·w=\{1+(-2),14+6,(-4)+5\} \\ (u\times v)·w=(1-2,14+6,-4+5) \\ (u\times v)·w=(-1,20,1) \end{gathered}$$

Step 4. Firstly finding the value of v×w

The cross product of

$v\times w=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 4& 0& 1\\ -2& 6& 5\end{array}\right]$

Now solving the cross product

role="math" localid="1649405232879" $$\begin{gathered} v\times w=\left(\right(0\left)\right(5)-(1\left)\right(6\left)\right)i+\left(\right(4\left)\right(5)-(1\left)\right(-2\left)\right)j+\left(\right(4\left)\right(6)-(0\left)\right(-2\left)\right)k \\ v\times w=(0-6)i+(20+2)j+(24+0)k \\ v\times w=-6i+22j+24k \end{gathered}$$

Step 5. Now finding the value of u·(v×w)

The vectors of$v\times w\mathrm{is}(-6,22,24)$

$$\begin{gathered} u·(v\times w)=(2,1,-3)·(-6,22,24) \\ u·(v\times w)=\{2+(-6),1+22,-3+24\} \\ u·(v\times w)=(-4,23,21) \end{gathered}$$