Q. 35 - Calculus

Question

In Exercises 30–35 compute the indicated quantities when $u=(-3,1,-4), v=(2,0,5), \mathrm{and} w=(1,3,13)$

Find the volume of the parallelepiped determined by the vectors u, v, and w.

Step-by-Step Solution

Answer

The volume of the parallelepiped determined by vectors u, v and w is $0$.

Step 1. Given Information

The indicated quantities when $u=(-3,1,-4), v=(2,0,5), \mathrm{and} w=(1,3,13)$

We have to find the volume of the parallelepiped determined by the vectors u, v, and w.

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

$$\begin{gathered} u·(v\times w)=\det \left[\begin{array}{ccc}-3& 1& -4\\ 2& 0& 5\\ 1& 3& 13\end{array}\right] \\ u·(v\times w)=-3\left|\begin{array}{cc}0& 5\\ 3& 13\end{array}\right|-1\left|\begin{array}{cc}2& 5\\ 1& 13\end{array}\right|-4\left|\begin{array}{cc}2& 0\\ 1& 3\end{array}\right| \\ u·(v\times w)=-3(0\times 13-5\times 3)-1(2\times 13-5\times 1)-4(2\times 3-0\times 1) \\ u·(v\times w)=-3(0-15)-1(26-5)-4(6-0) \\ u·(v\times w)=45-21-24 \\ u·(v\times w)=0 \end{gathered}$$