AAHL 3.15.2 Distance between a point and a line

00:00Hi guys, so in this

00:02lesson we're going to find

00:03the distance between a point

00:05and a line that might

00:06sound straightforward but it's actually

00:08kind of tricky. So these

00:11abs are finding the shortest

00:12distance between the point, this

00:13and the line, this. Okay,

00:15firstly, let's draw a line.

00:17If in any question, especially

00:19in vectors, if you can

00:20draw a diagram, it's going

00:23to help you. So here's

00:24a line, I don't have

00:25to draw it accurately in

00:27three dimensions or anything like

00:28that.

00:28That's just, I'm just gonna

00:29draw a line. And I'm

00:30gonna draw a point, there's

00:31a point here, P. So

00:34I need to find the

00:35distance, pre the point of

00:36the line. Now obviously, the

00:38shortest distance, it's not like,

00:40it's not like I'm gonna

00:41go, oh, that's the shortest

00:42distance. Clearly the shortest distance

00:44is, we need to go

00:45straight over, well let me

00:48start in the middle here.

00:49We need to go straight

00:50over to the line. Now

00:52what have I done there?

00:53Hopefully you can see that.

00:56This is perpendicular. These two

01:01lines are perpendicular because that

01:03would be the shortest distance.

01:05Imagine if you were the

01:07shortest distance between you and

01:09a wall, you just walk

01:10straight over to the wall,

01:11you would walk in a

01:12perpendicular line. Okay, so that's

01:15B. What's this point here

01:18that is on the line?

01:20Well, we don't know. Let's

01:23call it

01:24point A to begin with.

01:29Okay, now I set the

01:31word perpendicular there and I

01:33said before when you hear

01:35perpendicular in a vectors question,

01:38you should kind of immediately

01:39go, okay, the dot product

01:42must equal zero because that

01:44is what we have for

01:47perpendicular vectors, the dot product

01:49equals zero, but the dot

01:51product of what?

01:52So it would be the

01:53dot product of this vector

01:54either PA or AP. I'm

01:59actually going to go, I

02:00am going to go PA,

02:04will be a little bit

02:05easier to do. So we'll

02:06do PA dot the direction

02:09vector, this one, one, three,

02:12negative one. So this vector

02:13dot this vector must equal

02:16zero. So let's write this

02:17down here a second. So

02:18I'm going to write PA.

02:20pA, that vector dot the

02:24direction vector, 1, 3, negative

02:281. This dot, this has

02:30to equal 0. Okay, but

02:35what is pA? Let's find

02:37pA. So pA is A

02:42minus p, remember, I told

02:43you that. I always said

02:44that's very, very important. If

02:47not, go back and revise

02:48these are vectors, but PA

02:49is A minus P, where

02:50A and P are the

02:51position vectors of capital A

02:54and capital P respectively. So

02:56I have little P, I

02:58have the, because P, remember

03:01is two negative one, three,

03:03that's the point. So I

03:04have the position vector, the

03:05position vector is just two

03:07negative three, negative one, three

03:08as a vector. But what's

03:10the position vector of A?

03:13Well, the position vector of

03:14A is this,

03:16because it's some point, I

03:18don't know what it is

03:18yet, but it's some point

03:20on the line and all

03:21the points on the line

03:22have this position vector. So

03:24it's actually one plus one

03:27lambda, one plus lambda, zero

03:29plus three lambda. So that's

03:31just three lambda and two

03:33minus lambda, two minus lambda.

03:35That is the position vector

03:38of point A. And I'm

03:40going to do that minus

03:41P, which is two

03:44negative 1, 3. So this

03:48is PA, and this is

03:50going to be 1 plus

03:52lambda minus 2, which is

03:54minus 1 plus lambda, 3

03:57lambda plus 1, which is

03:593 lambda plus 1, and

04:012 minus lambda minus 3

04:03is minus 1 minus lambda.

04:09This is A minus P.

04:12or PA. So this dot

04:17this must equal zero. Let's

04:21bring that down here. Therefore,

04:25all right, therefore, therefore, minus

04:28one plus lambda, three lambda

04:31plus one, minus one, minus

04:35lambda, this dot this

04:40the direction vector 1, 3,

04:43negative 1 equals 0. Okay,

04:46let me re -explain why

04:49I'm doing this. So I

04:51want to find a lot,

04:54I want to find the

04:55point A. And I know

04:58that this vector PA, which

05:02ends up in this, and

05:03the direction vector are perpendicular,

05:05hence their dot product equals

05:070. Okay.

05:08Now, you can see here

05:10once I have a dot

05:11product equals zero and one

05:12variable, I'm going to be

05:14able to find lambda. Hence,

05:16I'm going to be able

05:17to find the position vector

05:19A, or I'll even be

05:22able to find the vector

05:23PA, which is this. This

05:26is PA. Okay, so this,

05:28the dot product of this

05:30times of this and this

05:31is this times one, which

05:33is minus one plus lambda,

05:36plus

05:36This times 3, which is

05:399 lambda plus 3, plus

05:43this times negative 1, which

05:45is 1 plus lambda negatives,

05:50negative 2. Negative times negative

05:52is positive, and this equals

05:530. So I'm left with

05:579 lambda, 10 lambda, 11

06:00lambda,

06:04equals negative one plus one

06:07is gone negative three. So

06:09lambda equals negative three over

06:1311. Fine, nearly there. Therefore,

06:20I have my lambda. This

06:22is PA. So let's do

06:23therefore again, therefore PA is

06:29equal to this with this

06:31for lambda. So I have

06:33minus 1 minus 3 1.

06:383 times 3 times minus

06:433 1. 3 times minus

06:473 1. And minus 1

06:53plus minus minus plus 3

06:571.

07:01And that's obviously simplified. This

07:03minus 1 minus 3 1

07:041 is minus 14 1

07:071 minus 11. Minus 11,

07:11minus 14 1 1. This

07:13will be minus 9 1

07:151 plus 1. Which is

07:192 1 and then finally

07:23minus 1 plus 3 1

07:261 will be minus

07:29is eight 11ths. And let's

07:34finish this off. The magnitude

07:37of PA, which is going

07:38to be the shortest distance

07:40I told you this wasn't

07:41easy. The shortest distance between

07:42the point and the line

07:43is going to be the

07:44magnitude of this. So it

07:50is 14 11th squared plus

07:56two.

07:5711th squared plus eight 11th

08:02squared and this I'm definitely

08:06Not gonna do it on

08:09my own I am I

08:11gonna use my calculator just

08:12while that is opening let's

08:14go back here and look

08:14at the question The shortest

08:17distance between the point and

08:19the line is going to

08:20be the distance of the

08:22vector PA hence why I've

08:24gone to all this trouble

08:25I could have found the

08:29point A using the same

08:30method, subbing it once I

08:32got lambda, I could have

08:32subbed it in here, and

08:34then found the distance between

08:35the two points. But what

08:36I'm doing is essentially exactly

08:38the same thing. So let

08:43me just move this here.

08:46I have a calculator, and

08:51I'm going to do the

08:53square root of the square

08:53root of the square root

08:53of the square root of

08:53the square root of the

08:53square root of 14 11th

08:58squared. So I need a

09:03bracket and I need this

09:06and I need 14 over

09:0811 and I need to

09:12close my bracket squared plus

09:17two 11ths. Let's just do

09:19it like this two 11ths.

09:21squared plus eight. Eight, 11th

09:32squared gives me 1 .4771

09:36equals 1 .4771. That's good

09:44enough for me and that's

09:46it. So yeah, certainly guys,

09:48I've seen that question and

09:49number

09:49times and past, past paper

09:51questions. It's, it, it actually,

09:53it's one of those that

09:54actually looks easier than it

09:56is. It's like shortest distance

09:57pre and point in line.

09:58That can't be hard, but

10:00it is because you have

10:01to go through all this,

10:02all this process. So hopefully

10:04that makes sense. Definitely practice

10:05one or two of those.

10:06Well, no, more than one

10:08or two, practice like seven

10:09or eight of them to

10:11make sure you know what

10:12you're doing. Okay, see you

10:13in the next lesson.