Parallel & perpendicular lines from graph

- [Instructor] In this video, we're gonna do a couple of examples that deal with parallel and perpendicular lines. So you have parallel, you have perpendicular, and of course you have lines that are neither parallel nor perpendicular. And just as a bit of a review, if you've never seen this before, parallel lines they never intersect. So let me draw some axes. So if those are my coordinate axes right there, it's my X-axis, that is my Y-axis, if this is a line that I'm drawing in magenta, a parallel line might look something like this. It's not the exact same line, but they have the exact same slope. If this moves a certain amount, if this change in Y over change in X is a certain amount this change change in Y over change in X is the same amount. And that's why they never intersect. So they have the same slope. Parallel lines have the same slope. Perpendicular lines, depending on how you wanna view it, they're kind of the opposite. Let's say this is some line, a line that is perpendicular to that, will not only intersect the line, it won't only intersect the line, it will intersect it at a right angle. At a 90 degree angle. At a 90 degree angle, and I'm not going to prove it for you here, I actually prove it in the linear algebra playlist, but a perpendicular lined slope, let's say this one right here, let's say that yellow line has a slope of M, then this orange line that's perpendicular to the yellow line is going to have a slope of negative one over M. Their slopes are gonna be the negative inverse of each other. Now, given this information, let's look at a bunch of lines and figure out if they're parallel, if they're perpendicular, or if they are neither. And to do that, we just have to keep looking at the slopes. So let's see, they say one line passes through the points four and negative three and negative eight, zero. Another line passes through the points negative one, negative one, and negative two, six. So let's figure out the slopes in each of these lines. I'll first do this one in pink. So this slope right here, so line one. So I'll call it slope one. Slope one is, let's just say it is, well I'll take this as the finishing point, so negative three minus zero, remember change in Y, negative three minus zero, over four, minus negative eight. Four minus negative eight, so this is equal to negative three, over, this is the same thing as four plus eight, negative three over 12, which is equal to negative 1/4. Divide the numerator by the denominator by three, that's this line, the first line. Now what about the second line? The second line, the slope for that second line is, well let's take, here negative one, negative one minus six, over negative one, minus negative two is equal to negative one minus six is negative seven, over negative one minus negative two, that's the same thing as negative one plus two, well that's just one. So the slope here is negative seven. So here their slopes are neither equal, so they're not parallel. Nor are they the negative inverse of each other, so this is neither, so this is neither parallel or perpendicular. Neither parallel nor perpendicular, so these two lines they intersect but they're not going to intersect at a 90 degree angle. Let's do a couple more of these. So I have here, once again, one line passing through these points, and then another line passing through these points. So let's just look at their slopes. So this one in green, what's the slope? The slope of the green one, I'll call that the first line. We could say, let's see change in Y, so we could do negative two minus 14, over, I did negative two first, so I'll do one first, over one minus negative three. So negative two minus 14 is negative 16. One minus negative three is the same thing as one plus three, that's over four, so this is negative four. Now what's the slope of that second line right there? So we have the slope of that second line, let's say five minus, so we'll say five minus negative three, that's our change in Y, over negative two minus zero. So this is equal to five minus negative three that's the same thing as five plus three, that's eight. And then negative two minus zero is negative two, so this is also equal to negative four. So these two lines are parallel. These two lines are parallel. They have the exact same slope. And I encourage you to find the equations of both of these lines and graph both of these lines and verify for yourself that they are indeed parallel. Let's do this one. Once again, this is just an exercise in finding slopes. So this first line has those points. Let's figure out it's slope. The slope of this first line. One line passes through these points, so let's say three minus negative three, that's our change in Y, over three minus negative six. So this is the same thing as three plus three which is six over three plus six which is nine. So the first line has a slope of 2/3. What is the second line's slope? This is the second line there. That's the other line passing through these points. So the other line's slope, let's see, we could say negative eight minus four, over two minus negative six. So what is this equal to? Negative eight minus four is negative 12. Two minus negative six, that's the same thing as two plus six, right? The negatives cancel out. So it's negative 12 over eight, which is the same thing if we divide the numerator and the denominator by four, that's negative 3/2. Notice these guys are the negative inverse of each other. If I take negative one over 2/3, that is equal to negative one times 3/2, which is equal to negative 3/2. These guys are the negative inverses of each other. You swap the numerator and denominator make 'em negative, and they become equal to each other, so these two lines are perpendicular. I encourage you to find the equations, I've already got the slopes for you, but find the equations of both of these lines plot them and verify for yourself that they are perpendicular.