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Are these points collinear?a) $A(-1,-2), B(3,2),$ and $C(5,5)$b) $D(a, c-d), E(b, c),$ and $F(2 b-a, c+d)$...

Question

Are these points collinear?a) $A(-1,-2), B(3,2),$ and $C(5,5)$b) $D(a, c-d), E(b, c),$ and $F(2 b-a, c+d)$

Are these points collinear? a) $A(-1,-2), B(3,2),$ and $C(5,5)$ b) $D(a, c-d), E(b, c),$ and $F(2 b-a, c+d)$



Answers

The points $A, B,$ and $C$ are collinear (in this order) if the relation $\|\overrightarrow{A B}\|+\|\overrightarrow{B C}\|=\|\overrightarrow{A C}\| \quad$ is satisfied. Show that $A(5,3,-1),B(-5,-3,1),$ and $C(-15,-9,3)$ are collinear points.

In this problem. Want to know the relation between points A. D and C. So a is over here. He's here and sees here. So question they asked us whether they're Khalidiya. The answer is they're not cleaner because they're not on the same line. However they do, they are on the same plane. So which means they're the points a D c R complainer. They are in the bass plane of just water. Uh, B is correct, but it's not.

Okay for a B. We know we can use the distance formula. So you have the screw of zero minus negative too. Three minus one squared equals to squirt or two for BC. We have squared of three minus. I know you have two squared six minus one squared equals five squirt of two on them for C A. You have the square root of three minus zero squared, plus six months, three, three squirt of two. So now, as you can see, a be people say sequels BC There for the answer to this is yes, and A is the one that lies between

And this question were asked to find the distance between A and B of the length of line segment A B, the length of line segment BC, and in the length of line segment A C also asked to determine if these points or Colin ear So I'm actually gonna start by plotting all three points a total of five negative five here that'll be point a point B is a 05 So that will be here and then point c is that point to one in my eyeball? That and look like to see that it looks like they're pretty Kalin ear. But let's just double check here. If we go from B to C, we go down. 1234 over, too. And then if we go down 1234 over. Two more Down 12 So it seems like we should be able to go down to over one down to over one, down to over one. We hit Ah, here, right. We say they're t connect B to C to take me to see we've got ah, slope here of down to over one. And so if we continue that to go from sea to aid down to over one, down to over one, down to over when we actually do hit see. So the slopes of those two components are those two pieces are the same. So it would appear that the points air Colin ear. So yes, they are Colin Ear. And we would say that see Point C is between A and B. All right, so that answers the second part of the question. Let's take a look at the first part of the question. The first part of the question asks us to find three distances. So we know that in general, are distance formula eyes equal to says that the distance between two points is equal to the square root of X two minus X one squared or the difference between the X coordinate squared and why two minus y one squared the difference between the Y coordinates square. So we're gonna take point a beast. Let's start with that points a to B and we're gonna say the distance there would be the square root of, ah, take the X coordinates the difference between the X coordinates, which would be five minus zero square that let me take the difference between the corresponding why Coordinates as well. So negative. Five minus five squared. So, essentially, I'm calling this X one. Why one on f? Rather, I called this X to y two on duh for point B. I called this X one y one Doesn't really matter which one's X one and why one a CZ Long as you're consistent in the directions off because I took the A coordinate minus toe be coordinate. I want to do that in both cases. So let's see how simplify here we're gonna have the square root of five minus zero is just five squared. Plus, here, we're gonna get negative. 10. But we want to remember that it's gonna be negative. 10 square. So we're gonna take negative 10 times. Negative. 10. This is a square root 25 plus 100. Or we can say that a B is the square root of 125. That's the distance there of a distance between B and C distance between B and C. We're gonna take again this time. We'll call this X two, and why, too? So we'll say, Let's see, what's the square root of Let's take the C coordinate, which is to minus the be coordinate and then the sea Coordinate to see why coordinate from point C and minus the y coordinate from point B and we'll square that. All right, so we have two minus zero, which is to swear one minus five that I would get me a negative four going to keep that in parentheses, because when I square it, I want to square the entire value. So I want to square the four and the negative. So two squared and the negative four times native for B positive 16. And so B C is the square root of 20. And there we have a B and B C. Last thing we need then is Ah, a c. So let's come over here. We'll do a C here and for a C. I'm gonna Let's see. Let's change this. We'll use this as X one. We'll make the A coordinates this time. The X one in the lie one and all right, so we're gonna take the square root for a C the square root of the X coordinate from see, which is to minus the X coordinate for A, which was five squared the Y coordinate from See, which is one minus a negative. Five squared. All right, so you've got to minus five. That'll be negative. Three squared, and then we have one minus and native, so it's gonna be plus five. So six square to hear. So be the square root of native three times a day. Three is 96 times six is 36. So here we've got the square root of 45 and that would be the value for a C.

Okay. Say we're looking at a bi first reason. A distant formal square of negative five months. Zero This is the square root of 41 squirt of 100 is 10. As you can see that some of any of these two lines are not equal to the length of the third line. Therefore, A B and C are non. They're not Colin ear.


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