Two perpendicular lines have the slop that multiplies to -1.
This is what the textbook and the teachers tell us, but rarely that they tell students why.
Here is my explanation that I found out when meditating in to this question.
There is another way to phrase the definition: the slop of two perpendicular lines are opposite reciprocals. Opposite numbers are numbers that have opposite signs. Reciprocals are numbers that multiplies to 1, or in other words flipped fractions.
For example:
2/5 × 5/2=1.
These two property combined gives the result of our numbers multiplying to -1.
Now we will separately look in to the property of opposite slops and slops that are reciprocals.
1. Opposite slop.
lines with opposite slops are reflected over the y-axis. When we think of slop as raise/ run, we can see that opposite slops goes to different directions with the same rate, when we think of the line as going left to right.
Example:
In the diagram to the right (01), the line with slop 3 and the line with slop -3 are reflection of each other over the y- axis (or the line x=0).
diagram-01
diagram-02
2. slops that are reciprocal
Slops that are reciprocals can be seen as flipped fractions. Flipping the amount of raise with the amount of run. This leads to the two lines to be reflected over the line y=x.
Example:
In the diagram to the left(02), the line with slop 1/3 and the line with the slop 3/1 (or 3), are reflection of each other over the line y=x.
When combining those two, we got a effect of first reflecting over y-axis than reflect over y=x, or first reflecting over y=x than reflect over y-axis. The sequence of action does not mater in this case.
Though I have not yet found an explanation for how combining two of these actions leads to a 90° rotation but here a example that will show how this works.
diagram-03
The red arrows in the diagram (04) show how a sector of the plain is transformed to a sector that is 90° rotation of the original.
diagram-04
And combining what I have explained. This will be the intuitive explanation for “why Two perpendicular lines have the slop that multiplies to -1.”
