# Slopes product = -1 <===> lines perpendicular

1. Sep 26, 2016

### Dank2

1. The problem statement, all variables and given/known data
Proof that if the slopes of two lines a1, a2 (that are not vertical), m1,m2 satisfy:
m1*m2 = -1, then the lines are perpendicular.

2. Relevant equations

3. The attempt at a solution
I tried to use the tan function, so that m1 = tanΘ where Θ1 is the angle of the line formed from x axis.
and m2 = tanΘ2.
now tanΘ21 = -1,
I tried using the tan(Θ1+Θ2) = but after simplifing i got (tanΘ1 - tankΘ2)/2
not sure if im on the right way, rather solve it using trig.

ok found the answer, i had to use the points (1,m1), (1, m2), form a triangle and check for Pythagorean theorem.

Last edited: Sep 26, 2016
2. Sep 26, 2016

### Staff: Mentor

Hint: You can use the addition theorem of cosine for $\Theta_1 - \Theta_2$.

3. Sep 26, 2016

### SammyS

Staff Emeritus
if you use the points, (1,m1), and (1, m2), then what's the third point you use to form the triangle?

How are these points related to the lines a1 and a2 and the slopes of these lines?

4. Sep 26, 2016

### Dank2

Yes, cos(Θ1-Θ2) = cosΘ1*cosΘ2 + sinΘ1*sinΘ2 = 1/sqrt(1+m12)* -1/sqrt(1 + (-1/m)2) + m1/sqrt(1+m12)*(-1/m1)/sqrt(1 + (-1/m)2) = 1 / * + -1 / * = 0 ==> Θ1-Θ2 = right angle, thanks. i see i must use the triangle to proof it in both ways.

Last edited: Sep 26, 2016
5. Sep 26, 2016

### Dank2

hey sammy, i've already seen a solution for it , and i marked the thread as solved. thanks.

6. Sep 26, 2016

### SammyS

Staff Emeritus
Well, it's true that I'm not grading your work, but if I were grading it, I would expect those issues to be addressed.

The points (1, m1) and (1, m2) are not likely to be on the lines involved .

7. Sep 26, 2016

### Dank2

Why not? If m1 = tanΘ =y/x
Then (x, m1x) is on the line and so (1, m1)

8. Sep 26, 2016

### Dank2

Maybe I should have added needed to be copied so that they will intersect with (0,0), and then we can always have the mentioned dots and form the triangle?

9. Sep 26, 2016

### SammyS

Staff Emeritus
... and perhaps:

The points (0, 0) and (1, m1) either lie on line a1 or lie on a line parallel to a1.

Similarly for (0, 0) and (1, m2) and line a2 ...

10. Sep 26, 2016

### Staff: Mentor

You don't need this as requirement. None of your lines is vertical. Thus you can always draw a coordinate system, in which the origin is the intersection point and, say the $x-$axis a third line (also intersecting at the origin of course). With that you have defined $\Theta_1$ and $\Theta_2$.

11. Sep 27, 2016

### mizanshoeb1

most probably you can use addition theorem of cosine for $\theta_1$ and $\theta_ 2$... i hope this hint will help you..

Last edited by a moderator: Sep 27, 2016