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
00:29is equal to negative one
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