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# Perpendicular lines from equation

Sal determines which pairs out of a few given linear equations are perpendicular. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

We are asked which of these
lines are perpendicular. And it has to be perpendicular
to one of the other lines, you can't be just perpendicular
by yourself. And perpendicular line, just
so you have a visualization for what for perpendicular lines
look like, two lines are perpendicular if they intersect
at right angles. So if this is one line right
there, a perpendicular line will look like this. A perpendicular line will
intersect it, but it won't just be any intersection,
it will intersect at right angles. So these two lines are
perpendicular. Now, if two lines are
perpendicular, if the slope of this orange line is m-- so let's
say its equation is y is equal to mx plus, let's say
it's b 1, so it's some y-intercept-- then the equation
of this yellow line, its slope is going to be the
negative inverse of this guy. This guy right here is going to
be y is equal to negative 1 over mx plus some other
y-intercept. Or another way to think about
it is if two lines are perpendicular, the product of
their slopes is going to be negative 1. And so you could write that
there. m times negative 1 over m, that's going to be-- these
two guys are going to cancel out-- that's going to be
equal to negative 1. So let's figure out the slopes
of each of these lines and figure out if any of them are
the negative inverse of any of the other ones. So line A, the slope is pretty
easy to figure out, it's already in slope-intercept
form, its slope is 3. So line A has a slope of 3. Line B, it's in standard form,
not too hard to put it in slope-intercept form, so
let's try to do it. So let's do line B over here. Line B, we have x plus 3y
is equal to negative 21. Let's subtract x from both sides
so that it ends up on the right-hand side. So we end up with 3y is equal
to negative x minus 21. And now let's divide both sides
of this equation by 3 and we get y is equal to
negative 1/3 x minus 7. So this character's slope
is negative 1/3. So here m is equal
to negative 1/3. So we already see they
are the negative inverse of each other. You take the inverse of 3, it's
1/3, and then it's the negative of that. Or you take the inverse of
negative 1/3, it's negative 3, and then this is the
negative of that. So these two lines are
definitely perpendicular. Let's see the third
line over here. So line C is 3x plus
y is equal to 10. If we subtract 3x from both
sides, we get y is equal to negative 3x plus 10. So our slope in this
case is negative 3. Now this guy's the negative of
that guy, this guy's slope is a negative, but not the negative
inverse, so it's not perpendicular. And this guy is the inverse of
that guy but not the negative inverse, so this guy is not
perpendicular to either of the other two, but line
A and line B are perpendicular to each other.