RWM102: Algebra

If you know the slopes of two non-vertical lines, can you easily determine if they are parallel? Perpendicular? Yes!

Slopes of Parallel Lines

Two lines in a plane are parallel if they never intersect; they have the same slant. We can use slope (or lack of it) to decide if lines are parallel:

Characterization of Parallel Lines Involving Slope

Suppose two distinct lines lie in the same coordinate plane.
Then, the two lines are parallel if and only if they are both vertical or they have the same slope.

This statement offers a good opportunity to review the mathematical words if and only if and or.

• The basic set-up is two lines lying in the same coordinate plane. This condition lurks in the background (is assumed to be true) for all that follows.
  
  
• Two mathematical sentences are being compared with the words if and only if:
  1. The two lines are parallel,
  2. They are both vertical OR they have the same slope.
     
     
• If and only if is a variation of is equivalent to. Sentences that are equivalent always have the same truth values – they are true at the same time, and false at the same time. If one is true, so is the other. If one is false, so is the other.
  
  
• An or sentence is true when at least one of the subsentences is true. The only time an or is false is when both subsentences are false.
  
  
• The nice, compact statement above is actually saying A LOT.

Slopes of Perpendicular Lines

Now, let's move on to perpendicular lines. Two lines are perpendicular if they intersect at a 90^ο angle. For example, the x-axis and y-axis are perpendicular.

Characterization of Perpendicular Lines Involving Slope

Suppose two non-vertical lines lie in the same coordinate plane. Let $m_{1}$ and $m_{2}$ denote the slopes of these lines.

The two lines are perpendicular if and only if their slopes are opposite reciprocals:

$m_{1}=-\frac{1}{m_{2}}$

Equivalently, the two lines are perpendicular if and only if their slopes multiply to 

$m_{1}m_{2}=-1$

Make sure you understand the opposite reciprocal equation:

$m_{1}=-\frac{1}{m_{2}}$

For example, what is the reciprocal of $2$? Answer: $\frac{1}{2}$

What is the opposite [of the] reciprocal of $2$? Answer: $-\frac{1}{2}$

• So, the numbers $2$ and $-\frac{1}{2}$ are opposite reciprocals.
• Also notice that $(2)(-\frac{1}{2})=-1$.
• So, lines with slopes $2$ and $-\frac{1}{2}$ are perpendicular.

It is easy to see that this is the correct characterization for perpendicular lines, by studying the sketch below:

The yellow triangle, with base of length $1$and right side of length $m$, shows that the slope of the first line is $\frac{rise}{run}=\frac{m}{1}=m$.

Now, imagine that this yellow triangle is a block of wood that is glued to the line. Rotate this block of wood counter-clockwise by 90^ο (so the original base is now vertical). Using the rotated triangle to compute the slope of the new, rotated, line gives:  $\frac{rise}{run}=\frac{1}{-m1}=-\frac{1}{m}$

Easy! Voila!

Source: Tree of Math, https://www.onemathematicalcat.org/algebra_book/online_problems/par_perpen.htm
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