Chapter 11.3, Problem 54E

To determine

To calculate: The scalar triple product for set of vectors $u=\langle 4,0,1\rangle ,v=\langle 0,5,0\rangle ,w=\langle 0,0,1\rangle$ .

Answer to Problem 54E

The scalar triple product is $20$ .

Explanation of Solution

Given information:

The set of vectors $u=\langle 4,0,1\rangle ,v=\langle 0,5,0\rangle ,w=\langle 0,0,1\rangle$ .

Formula used:

For the set of vectors, $u=u_{1}i+u_{2}j+u_{3}k$ , $v=v_{1}i+v_{2}j+v_{3}k$ and $w=w_{1}i+w_{2}j+w_{3}k$ , the cross scalar triple product is defined as ,

  $u\cdot \left(v\times w\right)=\begin{array}{ccc}{u}_1& u_2& u_3\\ v_1& v_2& v_3\\ w_1& w_2& w_3\end{array}$

Calculation:

Consider the set of vectors $u=\langle 4,0,1\rangle ,v=\langle 0,5,0\rangle ,w=\langle 0,0,1\rangle$ .

Recall that for the set of vectors, $u=u_{1}i+u_{2}j+u_{3}k$ , $v=v_{1}i+v_{2}j+v_{3}k$ and $w=w_{1}i+w_{2}j+w_{3}k$ , the cross scalar triple product is defined as ,

  $u\cdot \left(v\times w\right)=\begin{array}{ccc}{u}_1& u_2& u_3\\ v_1& v_2& v_3\\ w_1& w_2& w_3\end{array}$

Apply it for provided set of vectors,

  $\begin{array}{c}u\cdot \left(v\times w\right)=\begin{array}{ccc}4& 0& 1\\ 0& 5& 0\\ 0& 0& 1\end{array}\\ =4\begin{array}{cc}5& 0\\ 0& 1\end{array}-0\begin{array}{cc}0& 0\\ 0& 1\end{array}+1\begin{array}{cc}0& 5\\ 0& 0\end{array}\\ =4\left(5\right)-0\left(0\right)+1\left(0\right)\\ =20\end{array}$

Thus, the scalar triple product is $20$ .