Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the last lecture we saw how two linear equations
could describe parallel lines in the Cartesian plane.
Another possibility is that the lines described by two linear equations can be "perpendicular".
Perpendicular lines are lines which meet at right angles.
At their point of intersection, two perpendicular lines form four 90 degree angles.
Perpendicular lines don't have to be horizontal and vertical.
They may be oriented in any way as long as they meet at right angles.
So how can we determine if two linear equations describe perpendicular lines?
As we saw in the last lecture, parallel lines have the same slope.
Likewise, we can tell if two lines are perpendicular by comparing their slopes.
Perpendicular lines have slopes which are "negative reciprocals" of each other.
So what does it mean for slopes to be negative reciprocals?
We saw in pre-algebra, that multiplying two numbers that are reciprocals of each other
always produces a product of one.
Negative reciprocals can be created by changing the sign of one of the reciprocals.
Multiplying two numbers that are negative reciprocals
always produces a product of negative one.
So if the slopes of two intersecting lines are negative reciprocals of each other
the product of their slopes must be negative one
and this tells us that the two lines are perpendicular.
So why are the slopes of perpendicular lines negative reciprocals?
To see why this makes sense, let's draw a line running through the origin
and place a point on the line somewhere other than the origin.
The slope of this line can be calculated using any two points on the line.
So if we like, we can use the origin as one of the points.
The slope is equal to the difference in the two points' vertical positions
divided by the difference in their horizontal positions.
Travelling from the left point to the right point
let's call the difference in vertical positions "p"
and the difference in horizontal positions "q".
So the slope of the line is "p over q".
Now let's create a second copy of this diagram rotated 90 degrees counterclockwise.
Since the copy of the line is rotated from the original line by 90 degrees
the two lines are perpendicular.
For the second line, the slope can once again be calculated
from the difference in vertical positions divided by the difference in horizontal positions
as we travel from the left point to the right point.
This time, the difference in vertical positions is "negative q"
and the difference in horizontal positions is "p".
So the slope of the second line is "negative q over p".
These two slopes "p over q"
and "negative q over p"
are negative reciprocals of each other.
We can prove that these slopes are negative reciprocals by multiplying them together.
To multiply these two fractions
we multiply their numerators and denominators to create a single fraction.
Then moving the negative sign outside this fraction
and using the commutative law to swap q and p in the numerator
we see that the numerator and denominator of the fraction are the same.
So the fraction is equal to negative one.
These two slopes are therefore negative reciprocals of each other.
This is true for any two perpendicular lines
that is, unless one of the lines is vertical
since a vertical line's slope is undefined.
Using the fact that the slopes of perpendicular lines are negative reciprocals
allows us to construct an equation for a line which is perpendicular to any given line
and intersecting any given point.
For example, if we know that the given line's slope is one-half
then the slope of the perpendicular line must be the negative reciprocal of one-half
or negative two.
We can then use point-slope form
to write the equation of a perpendicular line through any given point.
For example, let's say we are given the equation of a line
and asked to write the equation for a perpendicular line
which runs through the point "four, negative five".
Since the equation for the given line is in slope-intercept form
the coefficient of x, "three"
corresponds to the constant m, the slope of the line.
Since the slope of the given line is three
the slope of the perpendicular line must be the negative reciprocal of three
or negative one-third.
Knowing the slope and the coordinates of a point on the line
we can write the equation for the perpendicular line using the point-slope form.
The constant m
is the slope, "negative one-third"
and constants x-one and y-one are the coordinates of the given point
four
and negative five.
The equation can then be simplified by writing "y minus negative five"
as "y plus five".
We have seen two geometric relationships
that can exist between lines described by linear equations.
In the next lecture, we will introduce a powerful mathematical concept
of finding solutions that simultaneously satisfy multiple equations.