91影视

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

We have to find the value of$(u脳v)路w\mathrm{and}u路(v脳w)$

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

Now solving the cross product.

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

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

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

The cross product of

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

Now solving the cross product

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

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