Perpendicular Lines
The theory behind perpendicular lines is similar to what we saw with parallel lines. Remember that, in geometry, two lines are perpendicular if they cross each other at a right angle. The algebra way of expressing this isn't quite as straightforward.
Two lines are perpendicular if they have their slopes are negative reciprocals of each other.
That "negative reciprocal" thing is a little confusing. What it means is that if you have a line then to get the slope of the line perpendicular to it you take the original line's slope, flip it over (that's the reciprocal part) and make it negative. Here's an easier way to think about it: If you multiply the slopes together and you get -1 then the lines are perpendicular.
Example 1
Are the lines $y = 2x + 4$ and $x = -3y - 2$ perpendicular?
To answer this question, we need to know the slopes of the two lines. The first equation is in slope-intercept form so we can immediately see that its slope is going to be 3. Be careful with the second equation - it may look like it's in the slope-intercept form but, if you look closely, you'll see that it's solved for x not y. If we solve it for y we get:
$$x = -3y - 2$$$$-3y = x + 2$$$$y = \frac{-1}{3}x - \frac{2}{3}$$
So the slope of the second line is -1/3. If we multiply the two slopes together we get:
That isn't equal to -1 so the lines aren't perpendicular.
Example 3
Are the lines $y=\frac{2}{3}x - 6$ and $y=-\frac{3}{2}x - 2$ perpendicular?
Both of these lines are already in the slope-intercept form so we can see right away that there slopes are 2/3 and -3/2. If we multiply those numbers together, we get:
$$\frac{2}{3}\cdot\frac{-3}{2}=\frac{-6}{6}=-1$$
Because the product equals -1, the lines must be perpendicular.
Example 2
What's the slope of the line perpendicalar to the line y = -4x + 3?
To find the slope of the perpendicular line we have to do two things: flip the orignal slope over and reverse its sign. The reciprocal of -4 is -1/4 and if we reverse its sign we get +1/4 so that's the slope of the perpendicular line.
Example 4
Find the equation of the line that's perpendicular to y = 2x + 4 that passes through the point (3, 5).
The procedure behind this kind of question goes like this: We know a point on the line so, if we know the line's slope, we can use the point-slope form of the equation of a line to find its equation. We can get the slope by taking the negative reciprocal of the slope of the that we're given.
Here's how it works in practice. We know that the slope of the original line is 2. That means that the slope of the perpendicular line is going to be -1/2. (Take a look at Example 2 to see how I worked that out.) Because the line passes through the point (3, 5), we can use the point-slope form of the equation of a line to get its equation.
$$y=-\frac{1}{2}(x-5)$$ $$y-6=-\frac{x}{2}+\frac{5}{2}$$ $$y=-\frac{x}{2}+\frac{5}{2} + 6$$ $$y=-\frac{x}{2}+\frac{5}{2} + \frac{12}{2}$$ $$y=-\frac{x}{2}+\frac{17}{2}$$
