Perpendicular lines

Geometry: Lines

Perpendicular lines

We have already seen that two perpendicular lines make an angle of $90^\circ$ or $\frac{\pi}{2}$ radians. This gives a relation between the slopes of two perpendicular lines.

Perpendicular Lines

For two lines $\blue k$ and $\green l$ with slope $a_{\blue k}$ and $a_{\green l}$ we have:

$\begin{array}{c} a_{\blue k} \cdot a_{\green l}=-1 \\ \text{ if and only if }\\ \text{line }\blue k \text{ and line } \green l \text{ are perpendicular}\end{array}$

This means that whenever $a_{\blue k} \cdot a_{\green l}=-1$ the lines are perpendicular.

And also that if the lines $\blue k$ and $\green l$ are perpendicular then $a_{\blue k} \cdot a_{\green l}=-1$ .

Given a line $k$ and a point $P$ we can use this to determine a line $l$ perpendicular to $k$ that passes through $P$ .

Determining Perpendicular Line

	Step-by-step	Example
	We determine the line $\green l$ perpendicular to a line $\blue k$ that passes through a point $P$ .	$\blue k: y=3x+5$ $P=\rv{3,2}$
Step 1	Determine the slope of line $\blue k$ .	$a_{\blue k}=3$
Step 2	Determine the slope of line $\green l$ using the rule $a_{\blue k} \cdot a_{\green l}=-1$ .	$a_{\green l}=-\frac{1}{3}$
Step 3	The equation of line $\green l$ is of the form $y=a_{\green l} \cdot x+b$	$\green l: y=-\frac{1}{3}x+b$
step 4	Determine $b$ by substituting the coordinates of point $P$ and solving the resulting equation.	$b=3$
step 5	Substitute $b$ in the equation from step 3.	$\green l: y=-\frac{1}{3}x+3$

In the graph the lines $l$ and $k$ from the example are displayed. We see that they indeed are perpendicular and that line $l$ passes through $P=\rv{3,2}$ .

Determine the equation of a line $l$ perpendicular to line $k: y=-{{8\cdot x}\over{9}}-{{1}\over{3}}$ through the point $P=\rv{-1, -2}$ .

$l:$ $y={{9\cdot x}\over{8}}-{{7}\over{8}}$

Step 1	We determine the slope of line $k: y=-{{8\cdot x}\over{9}}-{{1}\over{3}}$ . This is equal to $-{{8}\over{9}}$ .
Step 2	We now determine the slope of line $l$ with the rule: $a_k \cdot a_l=-1$ . This goes as follows: $\begin{array}{rcl}-{{8}\over{9}} \cdot a_l=-1 \\&&\phantom{xxx}\blue{\text{rule perpendicular lines with }a_k=-{{8}\over{9}}} \\ a_l=\frac{-1}{-{{8}\over{9}}} \\&&\phantom{xxx}\blue{\text{both sides divided by }-{{8}\over{9}}} \\ a_l={{9}\over{8}} \\&&\phantom{xxx}\blue{\text{simplified}} \end{array}$
Step 3	Line $l$ is of the form: $y={{9}\over{8}} \cdot x+b$ .
Step 4	We fill in point $P$ to determine $b$ . This gives the equation $-2={{9}\over{8}} \cdot -1+b$ We solve this linear equation for $b$ and find $b=-{{7}\over{8}}$ .
Stap 5	We fill in the $b$ we found in the equation from step $3$ . This gives: $l: y={{9\cdot x}\over{8}}-{{7}\over{8}}$

Remark that if you have written line $l$ in a different form that this is also correct, provided that the slope of the line is equal to ${{9}\over{8}}$ and the line goes through the point $P=\rv{-1, -2}$ .

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