IB · Mathematics Analysis and Approaches (AA)
Learn AASL 2.1.3 Parallel and perpendicular lines in this free IB Mathematics Analysis and Approaches (AA) video lesson for SL 2.1—Equations of straight lines, parallel and perpendicular.
Learn AASL 2.1.3 Parallel and perpendicular lines in this free IB Mathematics Analysis and Approaches (AA) video lesson for SL 2.1—Equations of straight lines, parallel and perpendicular.
The lesson focuses on parallel and perpendicular lines, emphasizing their relationship to gradient. Two lines are parallel if their gradients are equal, while they are perpendicular if the product of their gradients equals -1, expressed mathematically as:
To find the gradient of a line perpendicular to another, one can take the negative reciprocal of the original line's gradient. For example, if the gradient of line is , the gradient of the perpendicular line would be:
The lesson also includes examples of identifying whether lines are parallel, perpendicular, or neither based on their equations.
00:00In this lesson we're going
00:01to look at parallel and
00:03perpendicular lines. Now, parallel and
00:07perpendicular lines, it's all related
00:09to gradient. So here you'll
00:11see, this is what parallel
00:13means, hopefully we know what
00:14parallel means. Two lines are
00:16parallel if the gradients are
00:17equal. And two lines are
00:19perpendicular if they cross at
00:2290 degrees. Now the more
00:24technical definition is that
00:28The gradient of one line
00:30over the gradient of the
00:31other line. I'm going to
00:32explain what that means in
00:34a second. In terms of
00:37the equation of a line,
00:38you can see clearly here
00:40this y equals 3x plus
00:414 and this y equals
00:433x minus 7 are going
00:45to be parallel because the
00:463 is the gradient, the
00:50mx plus c, this, these
00:52gradients are the same. So
00:54I can tell straight away
00:55they're going to be parallel.
00:56lines. In this case, the
00:59three, this gradient is three,
01:02and this gradient is negative
01:04one -third. Now I can
01:06immediately see that those lines
01:08are going to be perpendicular
01:08because of this thing here.
01:11So you may hear the
01:14negative reciprocal, or you may
01:17hear flip the gradient and
01:22change the sign. So instead
01:24instead of 3 over 1,
01:25you have 1 over 3
01:26and you flip the sign.
01:27Here, instead of 17 over
01:295, the perpendicular gradient would
01:31be negative 5 over 17.
01:34And I'll do those five
01:34examples in a second, just
01:36want to show you how
01:36it works here. So, imagine
01:40I have a gradient like,
01:43let's go with this. So
01:45this gradient is 1, 2,
01:483, 4, 5, 6. So
01:49the gradient here, this m
01:51is
01:521 over 6 because the
01:54rise over the run is
01:561 over 6. Now if
01:57I was to draw the
01:58perpendicular line, let's draw the
02:01perpendicular line. Here, what's the
02:06perpendicular line? Well, it is
02:10there. That is perfectly perpendicular.
02:15How do I know that?
02:16Well, let's look at the
02:16gradient of this line.
02:20line is 1 across and
02:231, 2, 3, 4, 5,
02:256 down. So this gradient
02:26is negative 6. And you
02:29can see, you can actually
02:31visualize that as perpendicular. I'll
02:33do another example just to
02:35kind of show you how
02:35it works. Let's go with
02:40this as gradient of 3
02:43over 4. This is m
02:46m equals 3 over
02:48for the perpendicular gradient is
02:52going to equal here. It
02:59will be negative 4 over
03:013. And look, you can
03:02see here that I go
03:04along 3 and down 4.
03:08So this m would be
03:10negative 4 over 3. So
03:12that's what I say, the
03:13negative reciprocal. If a gradient
03:16into three quarters, the gradient
03:18of the perpendicular line is
03:20negative four over three. So
03:21flip this upside out, three
03:22over four becomes four over
03:23three and change the sign.
03:25Let's do a few examples
03:28here. So if this is,
03:31if a gradient is of
03:33a line is two, a
03:34gradient of L2, it says
03:36find the gradient of L2
03:37perpendicular to L1 with gradient
03:42and these are the different
03:42gradients. So basically what is
03:43the gradient of
03:44So I'm going to do
03:46M. I'm just going to
03:47put a little, that's not
03:50the perpendicular gradient. That's like
03:52perpendicular. Equals, so I'm going
03:54to do, if it's two,
03:56that's like two over one.
03:59Two is obviously equal to
04:00two over one. So if
04:01I flip two over one,
04:02I get one over two,
04:04but I have to make
04:04it negative. So it's negative
04:06one over two. Here, the
04:08perpendicular gradient is, if this
04:11is negative three over four,
04:12I flip it 4 over
04:143 and I'm going to
04:15change the sign but this
04:16is negative so this is
04:18positive. It's the opposite sign.
04:20Similarly here I'm going to
04:22flip it 5 over 17.
04:24I'm going to change the
04:25sign to negative. I'm going
04:28to flip it here to...
04:33So it's 9 so it's
04:34going to be 1 over
04:379 and it's going to
04:39be positive.
04:40Finally, I've given you quite
04:44a tricky one. This is
04:47the perpendicular gradient. So if
04:49you remember from my gradient
04:50video, if a gradient has,
04:53if the gradient is zero,
04:56then it's a horizontal line
04:57here. The perpendicular line would
05:00obviously be this line going
05:02straight down. But what's the
05:04gradient of a straight vertical
05:06line? We say it is
05:08this undefined doesn't have defined
05:13gradient. So I can't actually
05:15get the gradient of this
05:17line. Okay, let's do one
05:21more kind of example of
05:23how this can come up
05:24with equations of lines. So
05:27here I have, it's a
05:30state where the lines are
05:31parallel, perpendicular, are neither. So
05:34this one, these are two
05:36separate
05:36and I've arranged a bit
05:40slightly differently just so you
05:44see all the different types.
05:45But the key is to
05:45find the gradient. So what's
05:47the gradient in this line?
05:49Well, the gradient is negative
05:505. I might even say
05:52let's go M1 equals negative
05:555. M2, that would be
05:57the gradient of this line.
05:59It's negative 5 because it's
05:59y equals mx plus c.
06:02Now I deliberately put them
06:02the wrong way around to
06:03confuse you.
06:04to confuse it, but just
06:05to allow you to see
06:08that it's not always going
06:09to be the form like
06:10was Mx plus C. This
06:11is the form like was
06:12C plus Mx. So M
06:14is negative five. And this
06:16gradient is a fifth. Now
06:19what do we know about
06:20this? Negative five over one,
06:22positive one over five. This
06:23is perpendicular. So I'm going
06:25to say therefore, perpendicular. Second
06:30one. M1
06:33Well, I don't know straight
06:35away what it is. I
06:36don't know what m is.
06:37So I need to rearrange
06:38this equation. So let's do
06:40that. 2x plus 3y plus
06:425 equals 0. I need
06:45it in the form y
06:46equals mx plus c. So
06:48I'm going to do 3y
06:49equals negative 2x minus 5.
06:53I'm subtracting 2x. I'm subtracting
06:545. Y equals I'm going
06:57to divide everything by 3.
07:00So it's minus
07:012 over 3x or minus
07:032x over 3, that's the
07:04same thing, minus 5 over
07:063. What is the gradient
07:09here? Well, this gradient is
07:12negative 2 -thirds. So m1
07:16is negative 2 -thirds. And
07:19m2, look at this is
07:21my m2 here. It's also
07:23negative 2 -thirds. Therefore, this
07:26one is parallel.
07:29parallel, parallel, because they're the
07:32same gradient. And then the
07:33last one, right, this is
07:37in the form y minus
07:38y1 equals m into x
07:40minus x1, so our m
07:42is three here. And this
07:43is a form y equals
07:44mx plus c, so my
07:46m is two. What can
07:48I say, if this is
07:49gradient of three and this
07:49is a gradient of two,
07:51what is neither? That's not
07:53parallel, because they're different. And
07:55it's not perpendicular because it's
07:56not, it's not the
07:57negative or super cool to
07:58this is just neither. Okay,
08:02that's the lesson on parallel
08:05and perpendicular lines. We need
08:07to know what parallel and
08:09perpendicular means. We need to
08:10know how to find the
08:11gradient of a perpendicular line.
08:14And we need to be
08:15able to recognize from equations
08:19whether they are perpendicular or
08:22parallel or neither.
08:25you