In today's video we're looking at perpendicular lines, which are just lines that cross each other at 90 degrees. For example, this line of y equals 2x minus 2 is perpendicular to this line of y equals minus a half x plus 8 because they cross each other at exactly 90 degrees. Perpendicular lines will always cross each other like this. and sometimes the only way that you can be sure they're perpendicular is that they will have this little square thing that indicates that they meet at a right angle.
Another way you can tell that two lines are perpendicular though is that if you multiply their gradients together you'll get an answer of minus one, which we can show using the equation m1 times m2 equals minus one, where m1 is the gradient of the first line, and m2 is the gradient of the second line. For example, on our graph, the first line has a gradient of 2, so m1 equals 2, whereas the second line has a gradient of minus a half, so m2 is minus one half. So if we wanted to double check that these two lines were perpendicular, we could just multiply together their gradients, and see if it gives us minus 1. So if we give it a go, we'll just do 2 times minus a half, which indeed does give us minus 1. So these lines are definitely perpendicular. Let's try and use this same idea to try these two questions, where in both cases we're given two equations, and we need to find out whether the lines they represent are perpendicular or not. So for this first question, we can see that the gradients of our lines are 1 half and 3. So we know that m1 is a half, and m2 is 3. So to check if they're perpendicular, we need to multiply their gradients by doing 1 half times 3, which gives us 3 over 2, not minus 1. So we know that these two lines aren't perpendicular.
For this next one, we first of all need to change the 2y equals 3x plus 8 equation into the form of y equals mx plus c, because that will allow us to easily find its gradient. So to do that, we just divide the whole thing by 2 to get y equals 3 over 2x plus 4. And now that we've done that, we can tell that the gradients of our lines are 3 over 2 and minus 2 thirds. So to check if they're perpendicular, we just multiply these together by doing 3 over 2 times minus 2 thirds, and that gives us minus 6 over 6, which actually simplifies down to minus 1. So these two lines are perpendicular. The next thing we need to look at is how to find the gradient of a perpendicular line.
For example, if you were given this line, of y equals one third x plus three, and then told that this other line was perpendicular to it, you would need to be able to figure out this other line's gradients. For questions like this, we can use the same formula as we were using before, but this time we're effectively looking for the m2 value, because we want the gradient of the second line. So we're going to have to rearrange the equation first by dividing both sides by m1. so that we get m2 equals minus 1 over m1. Then in this case, because the gradient of our first line is one third, we know that m1 is one third, and so to find m2 we just do minus 1 over a third, which is minus 3, so the gradient of our perpendicular line is minus 3. Now sometimes you could also be asked to find the equation of that perpendicular line.
And for those questions they will normally give you one of the sets of coordinates that the line passes through. For example, if we were told that these two lines cross at, then we'd know both the gradient and the set of coordinates of this perpendicular line. And that's all the information that we need to work out the equation of a line.
We cover exactly how to find the equation of a line in another video though, so we won't cover that here. That's everything for this video, so if you enjoyed it then please do give us a like and subscribe, and we'll see you again soon!