Chapter 9.5, Problem 32E

a.

To determine

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

Answer to Problem 32E

  $\left(0\right)$

Explanation of Solution

Given information:

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

Calculation:

Consider the given vectors,

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

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

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

Now substitute these values in standard matrix,

  $\begin{array}{l}a\cdot \left(b\times c\right)=\left|\begin{array}{ccc}1& -1& 0\\ -1& 0& 1\\ 0& -1& 1\end{array}\right|\\ =1\left(0+1\right)+1\left(-1-0\right)+0\left(1-0\right)\\ =1-1\\ =0\end{array}$

Hence, the scalar triple product of vector $a\cdot \left(b\times c\right)$ is $0$ .

b.

To determine

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

Answer to Problem 32E

The vectors are co-planar.

Explanation of Solution

Given information:

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

Calculation:

The scalar triple product of vector $a=\langle 1,-1,0\rangle ,b=\langle -1,0,1\rangle ,c=\langle 0,-1,1\rangle$ is $\left(0\right)$

Hence, the vectors are co-planar.