Chapter 13.4, Problem 34WE

a.

To determine

To prove: Use slopes to show that $\frac{bd}{ac}=-1$ .

Explanation of Solution

Given information: Suppose two non-vertical vectors $\left(a,b\right)\text{ and }\left(c,d\right)$ are perpendicular.

Formula used: Slope $=\frac{\text{Change in }y}{\text{Change in }x}$ .

Proof:

two non-vertical vectors $\left(a,b\right)\text{ and }\left(c,d\right)$

Slope of $\left(a,b\right)$ $=\frac{\text{Change in }y}{\text{Change in }x}$

Slope of $\left(a,b\right)$ $=\frac{b}{a}$ .

Slope of $\left(c,d\right)$ $=\frac{\text{Change in }y}{\text{Change in }x}$

Slope of $\left(c,d\right)$ $=\frac{d}{c}$ .

two non-vertical vectors $\left(a,b\right)\text{ and }\left(c,d\right)$ are perpendicular.

Then, $\text{Slope of} \left(a,b\right)\times \text{Slope of }\left(c,d\right)=-1$ .

  $\begin{array}{l}\frac{b}{a}\times \frac{d}{c}=-1\\ \frac{bd}{ac}=-1\end{array}$

b.

To determine

To prove: Use slopes to show that $ac+bd=0$ .

Explanation of Solution

Given information: Suppose two non-vertical vectors $\left(a,b\right)\text{ and }\left(c,d\right)$ are perpendicular.

Formula used: Slope $=\frac{\text{Change in }y}{\text{Change in }x}$ .

Proof:

two non-vertical vectors $\left(a,b\right)\text{ and }\left(c,d\right)$

Slope of $\left(a,b\right)$ $=\frac{\text{Change in }y}{\text{Change in }x}$

Slope of $\left(a,b\right)$ $=\frac{b}{a}$ .

Slope of $\left(c,d\right)$ $=\frac{\text{Change in }y}{\text{Change in }x}$

Slope of $\left(c,d\right)$ $=\frac{d}{c}$ .

two non-vertical vectors $\left(a,b\right)\text{ and }\left(c,d\right)$ are perpendicular.

Then, $\text{Slope of} \left(a,b\right)\times \text{Slope of }\left(c,d\right)=-1$ .

  $\begin{array}{l}\frac{b}{a}\times \frac{d}{c}=-1\\ \frac{bd}{ac}=-1\\ ac+bd=0\end{array}$