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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.

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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.