Parallel and Perpendicular Lines


I recently talked about finding out the angle between two lines. A useful application of this formula is to determine whether two lines are parallel or perpendicular. Here’s how..

1. Parallel Lines

Suppose two lines are parallel. Then the angle between them must be 0. That is,  θ=0, which makes tanθ = 0.

Therefore, our formula tanθ = \(|\frac{m_1 – m_2}{1+m_1m_2}|\), leads us to m1 = m2.

In other words, the slopes of the two parallel lines must be equal. This seems obvious, as two parallel lines must make the same angle with a transversal (i.e. the X-axis)

Conversely, if the slopes of two lines are equal, then they must be parallel.

There’s one small result which you might want to remember:

Suppose the lines a1x+b1y+c1=0 and a2x+b2y+c2=0 are parallel.

Then the slopes of these lines must be equal. Therefore, we have -a1/b1 = -a2/b2 or a1/a2 = b1/b2

That is, the coefficients of x and y are proportional (the equations must be in the form as shown).

Okay. About perpendicular lines..

2. Perpendicular Lines

In this case θ = 90°, or cotθ = 0. Again, using our formula we have, \(|\frac{1+m_1m_2}{m_1 – m_2}|\) = 0

This leads us to m1m2=-1. That is, the product of the slopes of two perpendicular lines must be equal to -1

Conversely, if the product of the slopes of two lines equals -1, then the lines must be perpendicular.

And, analogous to the previous result, if the lines a1x+b1y+c1=0 and a2x+b2y+c2=0 are perpendicular, then -a1/b1 x -a2/b2 = -1, or a1a2 + b1b2 = 0.

That’s it for now. I’ll talk a bit more about equations of parallel and perpendicular lines while covering examples (so don’t miss them).

Lesson Summary

  1. Two lines, whose slopes are m1 and m2 are parallel if m1 = m2
  2. Two lines, whose slopes are m1 and m2 are perpendicular if m1.m2 = -1
  3. Two lines, whose equations are a1x + b1y + c= 0 and a2x + b2y + c= 0 parallel if a1/a2 = b1/b2
  4. Two lines, whose equations are a1x + b1y + c= 0 and a2x + b2y + c= 0 perpendicular if a1a2 + b1b2 = 0.

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