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$(u脳v)脳w\mathrm{and}u脳(v脳w)$

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.

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

Now finding the value of $(u脳v)脳w$

role="math" localid="1649401901776" $$\begin{gathered} (u脳v)脳w=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 1& 14& -4\\ -2& 6& 5\end{array}\right] \\ (u脳v)脳w=\left(\right(14\left)\right(5)鈭�(鈭�4\left)\right(6\left)\right)i+\left(\right(1\left)\right(5)鈭�(-4\left)\right(-2\left)\right)j+\left(\right(1\left)\right(6)鈭�(14\left)\right(-2\left)\right)k \\ (u脳v)脳w=(70+24)i+(5-8)j+(6+28)k \\ (u脳v)脳w=94i-3j+34k \end{gathered}$$

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

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

$$\begin{gathered} u脳(v脳w)=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 2& 1& -3\\ -6& 22& 24\end{array}\right] \\ u脳(v脳w)=\left(\right(1\left)\right(24)鈭�(鈭�3\left)\right(22\left)\right)i+\left(\right(2\left)\right(24)鈭�(-3\left)\right(-6\left)\right)j+\left(\right(2\left)\right(22)鈭�(1\left)\right(-6\left)\right)k \\ u脳(v脳w)=(24+66)i+(48-18)j+(44+6)k \\ u脳(v脳w)=90i-30j+50k \end{gathered}$$