Q. 29 - Calculus

Question

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

Find the volume of the parallelepiped determined by vectors u, v and w. Do u, v and w form a right-handed triple?

Step-by-Step Solution

Answer

The volume of the parallelepiped determined by vectors u, v and w $106 \mathrm{cube} \mathrm{unit}.$

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

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

Step 2. The volume of the parallelepiped determined by u, v, and w is the absolute value of the triple scalar product u · ( v × w ) .

Although we could first evaluate the cross product $v\times w$ 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 × 3 matrix formed from the components of u, v, and w as the rows.

Step 3. Thus, the required volume is

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

Step 4. The vectors u , v and w form a left-handed triple.

By Theorem 10.36, the vectors u, v and w form a left-handed triple, since the triple scalar product $u·(v\times w) = -106 < 0$.

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