Chapter 9.5, Problem 29E

a.

To determine

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

Answer to Problem 29E

  $\left(0\right)$

Explanation of Solution

Given information:

The given vectors are $a=\langle 1,2,3\rangle ,b=\langle -3,2,1\rangle ,c=\langle 0,8,10\rangle$ .

Calculation:

Consider the given vectors,

  $a=\langle 1,2,3\rangle ,b=\langle -3,2,1\rangle ,c=\langle 0,8,10\rangle$

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

  $\begin{array}{l}a\cdot \left(b\times c\right)=\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|\\ Wehavenvectors,\\ a_1=1,a_2=2,a_3=3\\ b_1=-3,b_2=2,b_3=1\\ c_1=0,c_2=8,c_3=10\end{array}$

Now substitute these values in standard matrix,

  $\begin{array}{l}a\cdot \left(b\times c\right)=\left|\begin{array}{ccc}1& 2& 3\\ -3& 2& 1\\ 0& 8& 10\end{array}\right|\\ =1\left(20-8\right)-2\left(-30-0\right)+3\left(-24-0\right)\\ =12+60-72\\ =0\end{array}$

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

b.

To determine

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

Answer to Problem 29E

vectors are coplanar.

Explanation of Solution

Given information:

The given vectors are $a=\langle 1,2,3\rangle ,b=\langle -3,2,1\rangle ,c=\langle 0,8,10\rangle$ .

Calculation:

The scalar triple product of vector $a=\langle 1,2,3\rangle ,b=\langle -3,2,1\rangle ,c=\langle 0,8,10\rangle$ is ZERO.

Hence, the vectors are coplanar.