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 volume of the parallelepiped determined by vectors u, v and wand using u, v and w form a right-handed triple.

Although we could first evaluate the cross product $v脳w$ and then take the dot product of the resulting vector with u,it is slightly more efficient to just take the absolute value of the determinant of the 3 脳 3 matrix formed from the components of u, v, and w as the rows.

$$\begin{gathered} \left|u路(v脳w)\right|=\left|\det \left[\begin{array}{ccc}2& 1& -3\\ 4& 0& 1\\ -2& 6& 5\end{array}\right]\right| \\ \left|u路(v脳w)\right|=\left|2\left|\begin{array}{cc}0& 1\\ 6& 5\end{array}\right|-1\left|\begin{array}{cc}4& 1\\ -2& 5\end{array}\right|+(-3)\left|\begin{array}{cc}4& 0\\ -2& 6\end{array}\right|\right| \\ \left|u路(v脳w)\right|=\left|2(0脳5-1脳6)-1\{4脳5-(-2)脳1\}+(-3)\{4脳6-(-2)脳0\}\right| \\ \left|u路(v脳w)\right|=\left|2(0-6)-1(20+2)+(-3)(24+0)\right| \\ \left|u路(v脳w)\right|=\left|2(-6)-1\left(22\right)+(-3)\left(24\right)\right| \\ \left|u路(v脳w)\right|=\left|-12-22-72\right| \\ \left|u路(v脳w)\right|=\left|-106\right| \\ \left|u路(v脳w)\right|=106 \end{gathered}$$

By Theorem 10.36, the vectors u, v and w form a left-handed triple, since the triple scalar product $u路(v脳w)=鈭�106<0$.

Hence, the vectors u, v and w does not form a right-handed triple.