Chapter 9.5, Problem 31E

a.

To determine

Find the scalar triple product of vector $a\cdot \left(b\times c\right)$ .

Answer to Problem 31E

  $\left(55\right)$

Explanation of Solution

Given information:

The given vectors are $a=\langle 2,3,-2\rangle ,b=\langle -1,4,0\rangle ,c=\langle 3,-1,3\rangle$ .

Calculation:

Consider the given vectors,

  $a=\langle 2,3,-2\rangle ,b=\langle -1,4,0\rangle ,c=\langle 3,-1,3\rangle$

The standard form scalar triple product of vectors $a,b,c$ is,

  $\begin{array}{l}a\cdot \left(b\times c\right)=\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|\\ Wehavenvectors,\\ a_1=2,a_2=3,a_3=-2\\ b_1=-1,b_2=4,b_3=0\\ c_1=3,c_2=-1,c_3=3\end{array}$

Now substitute these values in standard matrix,

  $\begin{array}{l}a\cdot \left(b\times c\right)=\left|\begin{array}{ccc}2& 3& -2\\ -1& 4& 0\\ 3& -1& 3\end{array}\right|\\ =2\left(12-0\right)-3\left(-3-0\right)-2\left(1-12\right)\\ =24+9+22\\ =\left(55\right)\end{array}$

Hence, the scalar triple product of vector $a\cdot \left(b\times c\right)$ is $\left(55\right)$ .

b.

To determine

Verify that vectors are coplanar. If not, find the volume of the parallelepiped.

Answer to Problem 31E

  $55cubicunit$

Explanation of Solution

Given information:

The given vectors are $a=\langle 2,3,-2\rangle ,b=\langle -1,4,0\rangle ,c=\langle 3,-1,3\rangle$ .

Calculation:

The scalar triple product of vector $a=\langle 2,3,-2\rangle ,b=\langle -1,4,0\rangle ,c=\langle 3,-1,3\rangle$ is $\left(55\right)$ so that the volume of the parallelepiped is equal to the magnitude of scalar triple product,

Hence, the volume of the parallelepiped is $55cubicunit$ .