Chapter 11.2, Problem 64E

To determine

To prove: That the dot product of two perpendicular vectors is zero.

Explanation of Solution

Calculation:

Consider that one of the vectors is $\overrightarrow{u}=\langle \overrightarrow{a},\overrightarrow{b}\rangle$ .

So, the slope of the vector $\overrightarrow{u}$ is $\frac{\overrightarrow{b}}{\overrightarrow{a}}$ .

Also, consider that the other perpendicular vector to the vector $\overrightarrow{u}$ be $\overrightarrow{v}$ .

The slope of the vector $\overrightarrow{v}$ is $-\frac{a}{b}$ .

Write the vector $\overrightarrow{v}$ using this slope as:

  $\overrightarrow{v}=\langle \overrightarrow{k}\overrightarrow{b},-\overrightarrow{k}\overrightarrow{a}\rangle$

The dot product of the above two vectors is,

  $\begin{array}{c}\overrightarrow{u}\cdot \overrightarrow{v}=\langle \overrightarrow{a},\overrightarrow{b}\rangle \langle \overrightarrow{k}\overrightarrow{b},-\overrightarrow{k}\overrightarrow{a}\rangle \\ =\overrightarrow{a}\cdot \left(\overrightarrow{k}\overrightarrow{b}\right)+\overrightarrow{b}\left(-\overrightarrow{k}\overrightarrow{a}\right)\\ =\overrightarrow{k}\overrightarrow{a}\overrightarrow{b}-\overrightarrow{k}\overrightarrow{a}\overrightarrow{b}\\ =0\end{array}$

It is seen that the dot product of the above two perpendicular vectors is 0.

Hence, the statement is proved.