Parallel and Perpendicular Lines

Parallel lines have the same gradient.

Equation of a straight line: y=\textcolor{#10a6f3}{m}x+c

\textcolor{#10a6f3}{m} \rightarrow gradient

c \rightarrow y- intercept

For lines to be parallel, the value of \textcolor{#10a6f3}{m} must be the same. 

Let’s take a look at the following lines:

y=\textcolor{#10a6f3}{2}x+1 (*)

y\textcolor{#10a6f3}{-2}x=5 (**)

y=\textcolor{#10a6f3}{2}x -4 (***)

The second equation (**) must be rearranged into the form

y=\textcolor{#10a6f3}{m}x+c , before comparing the gradients: 

y=\textcolor{#10a6f3}{2}x+5

These lines all have a gradient of \textcolor{#10a6f3}{2} \, (m=2) . These straight line equations are shown on the graph to the right.

Perpendicular lines meet at a right angle. When we multiply the gradients of a pair of perpendicular lines they multiply to \textcolor{#00d865}{-1}.

Another way of looking at this concept is to think of the perpendicular gradient as the ‘opposite sign and reciprocal’.

For instance, let’s say we have a straight line with a gradient: \textcolor{#00d865}{m=2}

Then the perpendicular gradient would be: \textcolor{#00d865}{m=-\dfrac{1}{2}}

\textcolor{#00d865}{2}\times\textcolor{#00d865}{-\dfrac{1}{2}}=\textcolor{#00d865}{-1}

Let’s take a look at the following lines:

y=\textcolor{#00d865}{3}x+1

y=\textcolor{#00d865}{-\dfrac{1}{3}}x+2

These lines are perpendicular as: \textcolor{#00d865}{3}\times\textcolor{#00d865}{-\dfrac{1}{3}}=\textcolor{#00d865}{-1}. These straight line equations are shown on the graph to the right.