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In Exercises 25-32, determine whether the given points are collinear: Points are collinear if they can be labeled P, Q, and R so that d(P,0) + d(Q,R) d(P,R): 25. (0...

Question

In Exercises 25-32, determine whether the given points are collinear: Points are collinear if they can be labeled P, Q, and R so that d(P,0) + d(Q,R) d(P,R): 25. (0,0), (1,2), (-1,-2) 26. (3,4), (0,0), (-3,-4)

In Exercises 25-32, determine whether the given points are collinear: Points are collinear if they can be labeled P, Q, and R so that d(P,0) + d(Q,R) d(P,R): 25. (0,0), (1,2), (-1,-2) 26. (3,4), (0,0), (-3,-4)



Answers

Testing for Collinear Points In Exercises $35-40$ ,
use a determinant to determine whether the points are
collinear.
$$\left(2,-\frac{1}{2}\right),(-4,4),(6,-3)$$

Okay, This question gives us these three points, and it wants us to determine if they're Colin Ear. So to do that, we just use our determinant. So we check if this area of the triangle determine it is zero or not. Because of this, determining the zero in the triangle has zero area. So all the points on the same line. So let's just do that. SAR determinant is well, the exes go in the first column, wise go in the second column and then ones go in the third column. So now let's just expand this term in and out going from column one. So this is equal to zero, minus the determinant of deleting the row and column 211.61 And then we're subtracting a one. Delete the row to let the column, because this to 12.4 and one. So now we just got to do these determinants out. So we got negative two minus 1.6 than minus two. Maya's 2.4. So this is equal to negative point for minus negative point for which is just point for minus point for 40 So since we have a determinant of zero. These points lie in the same line since this determinant that we checked equal zero. The points are colon ear.

I mean this question were given a couple of sets of points and were asked to tell if they are Colin Ear and if they are co when you're tell which which point is in the middle of the line. Um, so let's think about what it means for three points to be coat Lanier. Uh, and I can't really draw in three dimensions well, so we're going to use a two dimensional example, but it'll generalize into the third dimension if this is a point. P. Q and R. Uh, these air clearly not Cohen here because we can tell that because the line p Q or the Vector P Q. Is not parallel with the vector. Que are. If these were CO one year, right, if he had P que and under all this severe, they aren't evenly spaced to be a p Q and R. The Vector peak. You would be parallel to the Vectrix you are, And if these are parallel, great. If P Q is parallel to you are that means that the Vector P Q is equal to some skill or multiple of the vector. Q. R eso This is Yeah, um, this is how we'll tell if if 23 points or Colin here. If these vectors are parallel and even if we don't know which one is in the middle. Say we had you Q and R these they're supposed to be coping here. Let me just actually try a little bit better. Um, so we had P Q and R, and I don't know which one is in the middle. I can still use the same algorithm of comparing P Q. And I'm using different color. You are have drawn them a little offset so we can see them. But still, what we see is that, um, que is going to be parallel you draw. That symbol is still parallel to Q Are in this case, they're just anti parallel, so they're parallel but going in the opposite direction. Um, so we don't need to know which point is in the middle, are we? We could just check to make sure that P Q and Q are parallel, and if they are that that'll indicated they are all three points Arco veneer. And then from there we can just by examination, tell which point is in the middle. So it's concerted our first point. Hey, is gonna be He is 16 negative five. Hugh is 25 negative three and our is the 0.431 right for 31 So what we're gonna do is, um, make those make inspectors P Q and Q r and determine if they are parallel. So the vector p Q. Is going to be just the change between the point peon the point, you each of the components. So we look at the X component right between p and Q X increases by one. The changes one. In the y component, it decreases by one. And in the Z component, it increases by two. So this is our vector P Q. And I can move on to our electric. You are Director que are we compare again? The X values X increases by two. Why decreases by two and Z increases by four So we can see that these two vectors that can read it like this uh, P Q is just 1/2 of q r, right? If we multiplied Q. R record que are about what half would get p que? Um so this means we can express P que as just a scale multiple que are so p que is parallel to Q R. And remember, we showed that this means that P. Q and R. Cohen here two p q r. Uh, are I don't mean Teoh right like that. The points p Q and R R Co been here and that means that we can look and see which which of these points is gonna be in the middle. Um, and we can see that because, uh, if a point is gonna be in the middle, right of in a coma, in your sense, it's gonna be in the middle of a line. We can just, like, compare our X values. This is x one x three. The points in the middle will have the X value that's in the middle, same with the y value. And if we could make a draw well in three dimensions, we would see that that would be, um, true of the Z value as well eso we look and see that Q. It's X value is between one and four, right, So the X value of Q is between the X value of P and R. Same with no y. va. It's between six. And three. Um, so and we got to confirm we can busy about you. It's between negative five and one. So we know then that que is in the middle. Cool. So, um, this was one example on that we're still using this idea that if the vectors are parallel, that means that these points Air Colin year. So let's let's get more practice. With that, I will move on to another example. Easy. It will be here we have the 0.157 p point Q is 5 13 81 The point are, is 039 So again, what we're gonna do is go through these. Find the vectors P Q and Q R C. If they're parallel and if they are, then we can just examine these vectors to air these points to see which of the points is in the middle. If these are, in fact, Colin here, eso the vector p Q. Gonna look at the difference in exits for the difference and why That's eight and the difference in Z. That's a negative eight. So this is peak. You Ah, and Q are now drawing a different color you are is gonna be difference in X here. That's negative. Five difference in why is negative 10 and the difference and Z is positive. 10. So it may not be obvious that these are parallel, but we see that they are. And we can show that vice expressing peak you as four times the vector one too negative, too, right? I'm kind of factoring out of four here, um, to make this next conclusion a little bit easier to see, right? Same thing with Q are in fact are five here were all convicted of negative five, in fact. And we guess one to negative too, right? So it's still a, uh, In either case, it's going to be some scale multiple times the vector one too negative, too, right? So this means that I could represent p que then p Q is going to be, um, 4/5 q r. Right? If I divided que art by five and most, but by four, I would get peak you. So this means, of course, that p que is parallel the vector Q r. Uh, which means that p Q. R. R. Cohen here, right, um, showing that these are all parallel means that P. Q and R. R Colin here so I can write P. Q. Oh! Are our co one year? Um, knowing that their co linear means that we can just examine P Q and R to find which of these points is in the middle and by examination, receive that well, this Pete The X value for P is between exploit for Q and R. Same with the Y value right. Five is between 13 and three in the Z value. Seven is between negative one and nine. Right, so p in every direction. And every dimension is between Q and R. So we see that p is in the middle. Great. Let's move on, Teoh. Another example. See the example. See, now we have the points. He is 123 Que as too negative. 36 And our as, um, three Negative One night. Three. Beautiful. Great eso Let's do the same process, right? We're gonna find the vector P Q. We will get the differences between X coordinates. That's one difference. And why corn? It is negative. Five difference in Z Cornett is positive. Three right and that we q r. You are is going to be. The difference in the X value is one difference in the why value is positive to the difference in the Z. Value is three. Um, so now looking at this, we can see that there is no way that we could multiply Peak. You buy something to get, um, que are we? We know this because, um well, let me let me back up. Uh, if they were parallel, we would have P q equals some some scaler. A times the vector, Q R If they were if they were paddler parallel. We could write this for some scaler. Multiple A. So that would mean that P que, in this case in particular will be right in red. In this case in particular, would be a times one and a tense two and a times three. Right. That's scale multiplication. We're just multiplying each of our components by a right. So for a record, you are. We multiply each components about a so we should hopefully if their apparel will be able to represent a peak. You in this way. So me, uh, right, peach you down here and we can examine this equality we would have one negative. 53 Um So for our X value, we would have won equals eight times. One which means that a equals what? But we see that if we go to our why value we have negative five equals eight times. Two if a equals one, that means that Do we draw the air a little bit better on that? If a equals one, that means that negative five equals two. Uh, which is not true. The contradiction, right. So we know now that there is no such A that makes this expression true, you know, such day. Um and that means that these two vectors air not parallel So I could write that P Q is not parallel to Q R. Right. If these two are not parallel, that means that our vectors are not. There are points are not killing here. So P Q and R are not killing here. Cool. Um, and because of that co winning here, uh, it doesn't make sense for us to find a point in the middle because they make a triangle. And there's not a mid over Texas triangle. Um, what we want to our last example D the last example. D we have the points p 951 Q 11 18 4 and are 630 Right. Uh, we're gonna do exactly the same process, right? Three yet vectors. P Q. The mess difference and X is too difference in why Here is 13 on the difference in Z is three. And now our vic uric you are. This is going to be the difference in X is negative. Five. The difference in, um why is going to be negative? 15 and then the difference in Z is gonna be four negative for Excuse me. So, uh, here we have, um, these two vectors and we need again to see if they are parallel. Remember, if they were paralleled, would be able to write P Q equals a times Q r. Uh, for some rial A and A is not a vector. It is a real number. So it's just a moment of are the real numbers. And remember what that means. That means that the vector to 13 3 would have to equal the vector a times negative five a times negative 15 and a times negative, for we hope we have intuition that these are not going to be parallel. Um, and we can show that by finding a contradiction in this in this statement. So this would mean that two equals a times negative five, which would mean a equals negative 2/5. Right? Another statement that follows from a label. This statement star. Another statement that follows from star is that, um, 13 equals eight times negative. 15 right. Well, what I'm showing is that these X coordinates have to equal each other. These y cordons had to eat each other and easy accordance of each other. So because our white cordons had to eat each other, I know 13 must equal a times negative 15. Well, if any equals negative 15. And I plugged this in here, that would mean that 13 equals negative 2/5 times negative 15 which actually equals six. And that's not true, right? 13 is not equal. Six. Mr is not equal 13. So this is a contradiction. This is a contradiction. So, uh, there no such ex new, such a Sorry. No, Such a exists to make this statement true. Right? Um so no such exists because in such exists, we can't express peak. You as a scale multiple of Q R, which, of course, means that peak you is not parallel to Q R on these. That means that, uh, P Q and R. Are not Colin here. Cool. So, um, and all of these examples what we're doing is just trying to see if these two vectors that we can make between these points are parallel or not. Eso We construct two vectors between points P. Q and Q R. Or you could do P, R and Q R or, uh, any any combination of two vectors between three points as long as we're talking about two different vectors. If there Colin ear, those two vectors should be parallel. And if they're parallel, we should be able to express it like this. P. Q. Uh equals a some real number. Eight times you are, Um, And if we can't, we show that by finding a contradiction where, um, a is one thing in the X component in a different thing in the y component or the Z component or something like that. Um, And by doing that, we can, uh, prove that something is that prove it. Defectors air not parallel. And if they're not parallel. We know that the points, they're not cool in here. However, if they are parallel as, in example Air B making find some real number A such that PQ equals a times Q R. That would prove that these two vectors are parallel and our points are cool in here. And once we have that, we can just by examination toe which point is in the middle, because the X value of the middle point will be between the X values of, um, the end point. Same with the Y value scene with Z value.

In this video, I'll be going over applications of matrices and determinants specifically, how to use the determinant of a matrix to find if to determine if these three points lie on the same line over our Colin er so will create ourselves a little three by three matrix using these three points to create the first two columns, so we'll have to three, three and 3.5 and negative one and two. Well, then fill in this third column with all wants, and then you can use your favorite method for finding the determinant. The normal method is where you pick the number and cross out the rest of numbers and then find the determinant of the four numbers that would be left. I have a calculator that will do it for me that I have access to right now. But once you calculate the determinant, you'll find the determinant is one half, and so because that permanent is not equal to zero, that tells us that these points do not lie on the same line. They're not co linear. If it was zero, they would lie on the same line. Since it's not, they are not colon here

In this video, I'll be going over applications of matrices and determinants This question we're using Determinant of the Matrix to see if these to determine if these three points one in the same line or if they're co linear. So what we do is throw these points in to create the first two columns of a three by three matrix So zero to 12.4 and minus 11.6. Then there's third column we just fill in with once. Okay, so we then need to calculate the determinant of this matrix. If the determinant is zero, then that means they are on the same line. If it is not zero, then they do not lie on the same line. So you can use the normal, like, crossed out method, the normal cross out method where we have zero times the determinant of these four numbers, minus two times the determinant of those four numbers, plus one times the determinant of these four numbers. If you do all that math, you'll find the determinant does indeed come out to zero, meaning that these three points are cold in here.


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