Chapter 8.4, Problem 38E

a.

To determine

To find:The volume of the parallelepiped using the expression $\overrightarrow{p}.\left(\overrightarrow{q}\times \overrightarrow{r}\right)$ .

Answer to Problem 38E

The volume of parallelepiped is $-5$ .

Explanation of Solution

Given:

A parallelepiped is a prism whose opposite faces are parallelogram.

Calculation:

The volume of the parallelepiped can be calculated by using the expression $\overrightarrow{p}.\left(\overrightarrow{q}\times \overrightarrow{r}\right)$ .

Here,

  $\begin{array}{l}\overrightarrow{p}=\left(-3,1,5\right)\\ \overrightarrow{q}=\left(2,1,-4\right)\\ \overrightarrow{r}=\left(0,0,1\right)\end{array}$

Now,

  $\left(\overrightarrow{q}\times \overrightarrow{r}\right)=\left|\begin{array}{l}ijk\\ 21-4\\ 001\end{array}\right|$ .

Expand it along with $i,j,k$ respectively.

  $\begin{array}{c}\overrightarrow{q}\times \overrightarrow{r}=i\left(1\times 1-\left(-4\times 0\right)\right)-j\left(2\times 1-\left(-4\times 0\right)\right)+k\left(2\times 0-1\times 0\right)\\ =i-2j\end{array}$

Now,

  $\begin{array}{c}\overrightarrow{p}.\left(\overrightarrow{q}\times \overrightarrow{r}\right)=\left(-3,1,-5\right).\left(1,-2,0\right)\\ =\left(-3\times 1\right)+\left(1\times -2\right)+\left(-5\times 0\right)\\ =-5\end{array}$

Therefore, the volume of parallelepiped is $-5$ .

b.

To determine

To find: A $3\times 3$ matrix using the three vectors and calculate its determinant.

Answer to Problem 38E

The matrix is given below and the determinant of the matrix is same as the result we obtained in part a.

Explanation of Solution

Given:

A parallelepiped is a prism whose opposite faces are parallelogram.

Calculation:

A $3\times 3$ matrix can formed by writing the elements of three vectors in matrix form.

  $\left[\begin{array}{l}-315\\ 2\text{}\text{}1-4\\ 001\end{array}\right]$ .

Now, the determinant can be calculated by expanding the matrix along with the elements $-3,1,5$ respectively.

  $\begin{array}{c}\left[\begin{array}{l}-315\\ 2\text{}\text{}1-4\\ 001\end{array}\right]=\left(-3\right)\left|\begin{array}{l}1-4\\ 01\end{array}\right|-\left(1\right)\left|\begin{array}{l}2-4\\ 0\text{}1\end{array}\right|+\left(5\right)\left|\begin{array}{l}21\\ 00\end{array}\right|\\ =-3\left(1\times 1-\left(-4\times 0\right)\right)-1\left(2\times 1-\left(-4\times 0\right)\right)+5\left(2\times 0-0\times 1\right)\\ =-3i-2j\\ =-5\end{array}$

Therefore, the determinant of the matrix is same as the result we obtained in part a.