Why does the negative reciprocal of the slope tell us your new line is perpendicular?

Answer 1

Not sure of an exact proof, but here's a way to think about how it works: First note that the y axis is perpendicular to the x axis. Now take a line, say y = (2/3)*x. So we can take rise/run = 2/3, and from the origin, it rises +2 and runs +3, so that the point (3,2) is on the line and connect the origin to (3,2) and you have the line.

Now if you want to rotate this line counterclockwise 90°, to get a perpendicular line. Start at the origin and now we have an x' {that's x-prime, which is in the altered coordinate system}. Note that positive x' is 90° counterclockwise from regular x-axis (x' coincides with positive regular y-axis). And positive y' coincides with negative (regular x) axis.

So the point (x',y') which in our example (3',2') will rise' 2 on the y' axis (which is really moving -2 in the regular x direction). Then it will run' 3 on the x' axis (which is really moving +3 in the regular y direction). So we have the point (-2,3) is a point on the perpendicular line. Connect this to the origin and we have the perpendicular line, and note the negative reciprocal slope.