00:00Hi guys. So in this
00:03lesson, we're going to look
00:04at the parametric form and
00:05Cartesian equation of the line.
00:07So there's two more ways
00:08to write the equation of
00:09a line that we're going
00:11to look at. So we
00:11have our vector equation of
00:13the line. That's r equals
00:13a plus lambda b. This
00:15form here. So the first
00:18thing we want to do
00:18in this question says write
00:19this in parametric form and
00:21Cartesian form. So we need
00:22to be able to kind
00:23of move between the different
00:24forms very fairly easily. And
00:27I think
00:28If you understand how they
00:29work, that's pretty straightforward to
00:31do. So let's go with
00:33the parametric form first. Remember
00:35I said in the previous
00:36lesson, a nice trick is
00:38to think of, it's not
00:40a trick, it's actually what
00:42it is. Think of or
00:43as x, y, z. That's
00:46what or is, it's the
00:48general position vector, if you
00:50like, it's all the points,
00:53the position vectors of all
00:55the points.
00:56So this can just be
00:59written as 1 negative 2,
01:003 plus lambda 1, 5
01:04negative 4. Find the same
01:06thing, what is x, y,
01:07z. Now to write this
01:08in parametric form, all we're
01:10doing is we're saying x
01:11is equal to 1 plus
01:121 lambda. x is equal
01:15to 1 plus lambda, 1
01:18lambda. Then y is going
01:22to equal negative 2 plus
01:24plus 5 lambda. So y
01:27is equal to negative 2
01:30plus 5 lambda. And finally,
01:36z equals 3 minus 4
01:43lambda. And we're done. That's
01:45it. That's the parametric form
01:46of the equation of the
01:47line. So that's pretty straightforward.
01:50Once you have this, it's
01:51easy to put
01:52in this form. The Cartesian
01:54form is slightly more tricky.
01:57So, well it's not that
01:58hard, but remember, a straight
02:04line is an infinite number
02:06of points. It's a way
02:07of writing, or the equation
02:08of the line is a
02:09way of writing that line
02:10so we can connect the
02:12x -coordinate, y -coordinate, and
02:14z -coordinates. So you might
02:15try to come up with
02:19a way of
02:20writing with just x, y
02:22and z, like, well, this
02:24varies. Imagine you had x,
02:27not enough for this line,
02:28but imagine you had something
02:28like x plus 2y plus
02:303z equals 4. If you
02:34draw this in, well, let's
02:35do that right now in
02:36geodebra. We draw this in
02:38geodebra. What we get is,
02:47um,
02:48plus 2y plus 3z equals
02:544. This is actually the
02:58equation of a plane. And
03:00we're going to see this.
03:01We're going to study planes
03:03in more detail in the
03:05next few lessons, but that's
03:08the equation of a plane.
03:09So I can't write a
03:10Cartesian equation of a line
03:12like that. So what I
03:16So what we do is
03:19we say, right, well, we're
03:21going to actually, and this
03:22one is called the Cartesian
03:23equations of a line because
03:25there's a two -equal signs
03:27for like the way we're
03:29going to get this is
03:30we're going to make lambda
03:31the subject in each of
03:33these little equations. So let's
03:36call this a, this is
03:37the parametric form, a parametric
03:40form, b Cartesian form. So
03:43in Cartesian form,
03:44I'm going to make lambda
03:46the subject here. So lambda
03:48is equal to x minus
03:511. And then here, I'm
03:54going to make lambda the
03:55subject. Lambda equals y plus
03:592 at the 2 divided
04:01by 5. And here, I'm
04:03going to say lambda equals
04:07z minus 3 divided by
04:11negative 4.
04:12Now if these are all
04:13equal to lambda, then these
04:15are all equal. And I
04:16can say x minus 1
04:19equals y plus 2 over
04:245 equals z minus 3
04:28over negative 4. So this
04:30is the Cartesian equation, Cartesian
04:35equations of this line. And
04:38this is another way of
04:39writing this
04:40exact thing here. And this
04:43will actually, if I have
04:44a point, so let's say
04:46I have a point that's
04:47on the line, like let's
04:49make up a point, two,
04:51let's say two, one, zero.
04:54If this point is on
04:56this line, then when I
04:57sub in two for x,
04:59one for y, and zero
05:01for z, this will equal
05:03this and it will equal
05:04this. It will satisfy the
05:06equations. However, if I sub
05:08this and I'm going
05:08I get two minus one
05:09is one. If I sub
05:11one in here, I get
05:12three divided by five, which
05:13is three -fifths. And if
05:15I sub in zero, I
05:16get three -quarters. Three -fifths
05:17does not equal three -quarters,
05:19and it does not equal
05:19one. So this point is
05:21definitely not on the line.
05:25Okay, example two. This time,
05:30I'm going to, well, I've
05:31given you the same equations
05:34of alignment, the Cartesian equations
05:36of alignment.
05:36and we're gonna go backwards.
05:38So we're gonna write this
05:39as a vector equation of
05:41the line now. How do
05:43I do it? Well, as
05:44I said, we're just gonna
05:45go backwards. We're gonna set
05:46this equal to, we're gonna
05:49set this equal to lambda.
05:51So I have x minus
05:53one over one if you
05:55like, equals lambda. Come up.
06:00I have y plus two
06:03over five equals
06:04lambda and I have z
06:08minus three over negative four
06:11equals lambda. And then I'm
06:15just going to write it
06:16x is equal to one
06:18times lambda is lambda plus
06:20one lambda plus one. Come
06:23on. Why equals multiply by
06:26five, five lambda minus two
06:29and then z
06:33equals negative 4 lambda plus
06:373. If we go back
06:38to this here, do we
06:41have the same thing? X
06:45equals 1 plus lambda minus
06:462 plus 5 lambda and
06:483 minus 4 lambda. Yes,
06:50we do and yet it's
06:50probably better that I do
06:51write it like that 1
06:53plus lambda. Y equals negative
07:002 plus
07:01So 5 lambda and z
07:04equals 3 minus 4 lambda.
07:07So this is my parametric
07:09form and then I can
07:11change this into or equals
07:151. So I have my
07:171 negative 2 and 3.
07:20That's my position vector 1
07:22negative 2 and 3. And
07:23then I'm going to do
07:24plus lambda times 1, 5
07:28and negative 4.
07:29or one, five, negative four.
07:31There we go. So that's
07:32going from Cartesian form into
07:37vector equation form. Easy, and
07:40this way is going from
07:41vector equation form into parametric
07:44form and Cartesian form. Now,
07:47last thing I'll say is
07:48when it's in this form,
07:50one of the nice things
07:50about this is you can
07:52kind of see straight away
07:53the position vector like this
07:55form, the position vector
07:57is M1, negative 2 and
08:013 and the direction vector
08:02is 1, 5 and negative
08:034 so you can kind
08:04of read it off straight
08:05away. I still recommend going
08:07through this way instead of
08:08reading it off. If you
08:10have a situation where 0,
08:11the direction vector has a
08:130 in it, then you
08:16won't be able to just
08:17read it off nicely. You
08:19won't have a 0 underneath
08:21the line here. You would
08:23just have
08:25Let me give you one
08:30example. So imagine we had
08:33r equals, let's just go
08:35one, two, three. Let's call
08:37this example three. So imagine
08:42we had not one over
08:44two. Imagine we had r
08:47equals one, two, three plus
08:51plus.
08:53Let's go with t times.
08:57Let's say we had two
09:00zero one. Now what we
09:03do is we say x
09:07equals one plus two t.
09:10y equals two plus zero
09:14t, which is just two,
09:16and then z equals three
09:19plus.
09:21t. Make t the subject.
09:28So I'm going to have
09:29t is equal to x
09:31minus 1 over 2. Here,
09:35t is equal to y
09:37minus 2 over 0. So
09:39we've got a problem. And
09:41then here, i t equals
09:43z minus 3. So in
09:46this situation, there is no
09:47t. I can't, I don't
09:49have a
09:49t because it's there is
09:51zero t. So the only
09:53way I can write this
09:54equation is I can say
09:56x minus 1 over 2.
10:01x minus 1 over 2
10:02equals z minus 3 over
10:081. And then just you
10:10have to do comma y
10:13equals 2. That's the only
10:14way to do it. You
10:15can't.
10:17you can't put an equal
10:18sign here because y is
10:20always equal to 2 which
10:21just won't satisfy the other
10:22two equations. Okay, that's it.
10:25Hopefully that makes sense. I
10:26definitely want you and need
10:28you to be able to
10:29kind of go between happily
10:32go between the three and
10:33be confident and comfortable with
10:35the three. So in you
10:36could get a question where
10:37they give you this that
10:39you can get a question
10:39where they give you this
10:40you can get a question
10:41where they give you this
10:42and I don't just mean
10:43a question where they say
10:44turn this into this or
10:45turn this into this. I
10:47mean a question of some
10:48type where you're just dealing
10:50with this so you need
10:52to be comfortable enough with
10:55this form and this form,
10:57this form that you can
10:58solve whatever the problem that
11:01you are given without worrying
11:04too much about what form
11:08the equation of line is
11:09in. Okay, hopefully that makes
11:11sense and I'll see you
11:12in the next lesson.
11:13you