Parallel & perpendicular lines from graph

In this video, we're going to 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.

Just as a bit of a review, if you've never seen this before, parallel lines never intersect. So if, let me draw some axes. So if those are my coordinate axes right there, that'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 in y over change in x is the same amount. And that's why they never intersect; they have the same slope. Parallel lines have the same slope.

Perpendicular lines, depending on how you want to view it, they're kind of the opposite. If, let's say that 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. And I'm not going to prove it for you here; I'll actually prove it in the linear algebra playlist.

But a perpendicular line's slope, so let's say that 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 1 over m. Their slopes are going to 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 (4, -3) and (-8, 0). Another line passes through the points (-1, -1) and (-2, 6). So let's figure out the slopes of each of these lines.

I'll first do this one in pink. So this slope right here—so line one—I'll call it slope one. Slope one is, let's just say it is, well, let's take this as the finishing point. So -3 minus 0, remember change in y. -3 minus 0 over 4 over 4 minus -8. So this is equal to -3 over—this is the same thing as 4 plus 8— -3 over 12, which is equal to -1/4. Divide the numerator and denominator by 3, that's this line—that's the first line.

Now, what about the second line? The second line's slope for that second line is, let's take here -1, -1 minus 6 over -1 minus -2. -1 minus -2 is equal to -1 minus 6, which is -7 over -1 minus -2. That's the same thing as -1 plus 2. Well, that's just 1.

So the slope here is -7. 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 parallel nor 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 -2 minus 14 over I did -2 first, so I'll do 1 first over 1 minus -3.

So -2 minus 14 is -16. 1 minus -3 is the same thing as 1 plus 3. That's over 4. So this is -4. Now what's the slope of that second line right there? So we have the slope of that second line: let's say 5 minus—so say 5 minus -3, 5 minus -3—that's our change in y over -2, -2 minus 0.

So this is equal to 5 minus -3, that's the same thing as 5 plus 3; that's 8, and then -2 minus 0 is -2. So this is also equal to -4. So these two lines are parallel. These two lines are parallel; they have the exact same slope.

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—it's just an exercise in finding slopes.

So this first line has those points. Let's figure out its slope. The slope of this first line—one line passes through these points—so see (3, 3) minus (-3, -3) that's our change in y over 3 minus -6.

So this is the same thing as 3 plus 3, which is 6 over 3 plus 6, which is 9. So this first line has a slope of 2/3. What is the second line's slope? So 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 -8, -8 minus 4 over 2 minus -6. So what does this equal to? -8 minus 4 is -12; 2 minus -6, that's the same thing as 2 plus 6.

Alright, the negatives cancel out. So it's -12 over 8, which is the same thing if we divide the numerator and denominator by 4; that's -3/2. Notice, notice these guys are the negative inverse of each other. If I take -1 over 2/3, that is equal to -1 times 3/2, which is equal to -3/2.

These guys are the negative inverses of each other. You swap the numerator and denominator, make them negative, and they become equal to each other. So these two lines are perpendicular. Perpendicular.

I encourage you to find the equations. I 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.