Chapter 10.2, Problem 6E

To determine

To find: The dot product of two vectors that are orthogonal.

Answer to Problem 6E

The dot product of two orthogonal vectors is $0$ .

Explanation of Solution

Given information:

Both the vectors are orthogonal.

Calculation:

For any two vector $u$ and $v$ , the dot product is given by $u\cdot v=\Vert u\Vert \Vert v\Vert \cos \theta$ , where $\theta$ is the angle between both the vectors.

Substitute $\theta =\frac{\pi }{2}$ as both the vectors are orthogonal.

  $u\cdot v=\Vert u\Vert \Vert v\Vert \cos \frac{\pi }{2}$

Substitute $\cos \frac{\pi }{2}=0$ in the above equation.

  $\begin{array}{c}u\cdot v=\Vert u\Vert \Vert v\Vert \cos \frac{\pi }{2}\\ =0\end{array}$

Therefore, the dot product of any two orthogonal vectors is $0$ .