00:01
In this question, we're given a couple sets of points, and we're asked to tell if they are co -linear.
00:06
And if they are co -linear, tell which point is in the middle of the line.
00:10
So let's think about what it means for three points to be co -linear.
00:16
And i can't really draw in three dimensions well.
00:20
So we're going to have to use a two -dimensional example, but it'll generalize into the third dimension.
00:25
If this is our point p, q, and r, these are clearly not co -linear, because, and we can tell that because the line p -q or the vector p -q is not parallel with the vector qr.
00:42
If these were co -linear, right, if we had p -q, and i'll draw this so that they aren't evenly spaced, we had p, q, and r, the vector p -q and r, the vector p -q and r, the vector p -q, would be parallel to the vector qr.
00:59
And if these are parallel, right, if pq is parallel to qr, that means that the vector pq is equal to some scale or multiple of the vector qr.
01:19
So this is, yeah, this is how we'll tell if two, if three points are co -linear, if these are parallel.
01:30
And even if we don't know which one is in the middle, say we had pq and r, these are supposed to be co -linear.
01:38
Let me just actually draw it a little bit better.
01:41
So we had p, q, and r, and i don't know which one is in the middle.
01:48
I can still use the same algorithm of comparing pq and let me use a different color, qr.
01:58
I've drawn them a little offset so we can see them.
02:00
But still, what we see is that, pq is going to be parallel, let me draw that symbol, is still parallel to qr.
02:13
In this case, they're just anti -parallel, so they're parallel, but going in the opposite direction.
02:20
So we don't need to know which point is in the middle.
02:22
We can just check to make sure that pq and q are parallel, and if they are, that'll indicate that they are, all three points are co -linear.
02:32
And then from there we can just by examination tell which point is in the middle.
02:38
So let's get started.
02:40
Our first point a is going to be he is 1 -6 -9 -5, q is 2 -5 -9 -3, and r is the point 4 -3 -1.
03:00
4 -3 -1.
03:02
So what we're going to do is make those.
03:06
Make those vectors pq and qr and determine if they are parallel.
03:12
So the vector pq is going to be just the change between the point p and the point q in each of the components.
03:20
So we look at the x component, right? between p and q, x increases by one.
03:24
The change is one.
03:26
In the y component, it decreases by one.
03:29
And in the z component, it increases by two.
03:32
So this is our vector pq.
03:35
And now i can move on to our vector q.
03:37
Or vector qr, we compare again the x values, x increases by 2, y decreases by 2, and z increases by 4.
03:55
So we can see that these two vectors, i can write it like this, pq, is just one -half of qr.
04:10
Right if we multiplied qr the vector qr by one half we would get pq so this means we can express pq as just a scalar multiple qr so pq is parallel to qr and remember we show that this means that pq and r are co -linear so pqr i don't i don't mean to write like that the points pq and r are co -linear.
04:45
And that means that we can look and see which, which of these points is going to be in the middle.
04:52
And we can see that because if a point is going to be in the middle, right, in a co -linear sense, it's going to be in the middle of a line.
05:05
We can just compare our x values.
05:07
This is x1, x3.
05:09
The point that's in the middle will have the x value that's in, the middle, same with the y value.
05:15
And if we could, if i could draw well in three dimensions, we would see that that would be true of the z value as well.
05:24
So we look and see that q, its x value, is between 1 and 4, right? so the x value of q is between the x value of p and r.
05:36
Same with the y value.
05:38
It's between 6 and 3.
05:41
And we can also confirm looking at the z value.
05:44
It's between negative 5 and 1.
05:47
So we know then that q is in the middle.
05:56
Cool.
05:57
So this was one example, and that we're still using this idea that if the vectors are parallel, that means that these points are co -linear.
06:07
So let's get more practice with that.
06:09
And we'll move on to another example.
06:11
Example b here, we have the point 157.
06:17
It's p.
06:19
The point q is 513 negative 1, and the point r is 039.
06:32
So again, what we're going to do is go through these, find the vectors pq and qr, see if they're parallel, and if they are, then we can just examine these vectors to, or these points to see which of the points is in the middle, if these are, in fact, collinear.
06:52
So the vector pq, we're going to look at the difference in x, that's four, the difference in y, that's eight, and the difference in z, that's negative 8.
07:07
So this is pq, and qr now, let me draw it in a different color, qr, is going to be difference in x here, that's negative 5, difference in y, is negative 10 and the difference in z is positive 10.
07:35
So it may not be obvious that these are parallel, but we see that they are, and we can show that by expressing pq as 4 times the vector 1 to negative 2, right? i'm kind of factoring out of 4 here to make this next conclusion a little bit easier to see.
07:58
Right.
07:59
Same thing with qr, i can factor out of five here, or i can factor out of negative five, in fact.
08:04
And we get 1, 2, negative 2.
08:10
So it's still a, in either case, it's going to be some scale of multiple times the vector 1, 2, negative 2.
08:19
Right? so this means that i could represent pq.
08:26
Then pq is going to be, 4 5ths qr.
08:36
If i divided qr by 5 and multiplied it by 4, i would get pq.
08:41
So this means, of course, that pq is parallel to the vector qr, which means that pqr are co -linear, right? showing that these are all parallel means that pq and r are co -linear.
09:00
So i can write p, q, r are a co -linear.
09:08
Knowing that they're co -linear means that we can just examine pq and r to find which of these points is in the middle.
09:17
And by examination, we see that, well, this, the x value for p is between the x value for q and r.
09:26
Same with the y value, right? five is between 13 and 3, and the z value 7 is between negative 1 and 9.
09:34
Right so p in every direction in every dimension is between q and r so we see that p is in the middle great let's move on to another example c the example c now we have the points p is one two three q as two negative three six and r as um three negative one 9 .3, negative 1.
10:19
Great.
10:19
So let's do the same process.
10:21
We're going to find the vector pq.
10:23
We look at the differences between the x coordinates.
10:26
That's one...