Chapter 13.2, Problem 44E

To determine

To find: The cross product of vectors $\overrightarrow{v}=\langle 2,0,3\rangle$ , and $\overrightarrow{w}=\langle 2,5,0\rangle$ , and verify that the resulting vector is perpendicular to $\overrightarrow{v}$ , and $\overrightarrow{w}$ .

Answer to Problem 44E

The cross product of the vector is $\langle -15,6,10\rangle$ , and vectors are not perpendicular.

Explanation of Solution

Given:

The vectors are $\overrightarrow{v}=\langle 2,0,3\rangle$ , and $\overrightarrow{w}=\langle 2,5,0\rangle$ .

Calculation:

Find the cross product of the vectors.

  $\begin{array}{c}\overrightarrow{v}\times \overrightarrow{w}=\left|\begin{array}{ccc}i& j& k\\ 2& 0& 3\\ 2& 5& 0\end{array}\right|\\ =i\left(0-15\right)-j\left(0-6\right)+k\left(10-0\right)\\ =-15i+6j+10k\\ =\langle -15,6,10\rangle \end{array}$

The vectors will be perpendicular if their dot product is zero.

Find the dot product of the given vectors.

  $\begin{array}{c}\overrightarrow{v}\cdot \overrightarrow{w}=\langle 2,0,3\rangle \cdot \langle 2,5,0\rangle \\ =4+0+0\\ =4\end{array}$

Therefore, the cross product of the vector is $\langle -15,6,10\rangle$ , and vectors are not perpendicular.