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Do the points A(2,1,5), b(-1,-1,10), and c(8,5,-5) define a plane? Explain why or why not.

Mathematics
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  • phi
use elimination to find if the three points are independent (not collinear)
Elimination? -I think no, because there is no position vector or a direction vector.
Pi?

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Other answers:

@Phi What is elimination?
  • phi
http://www.wolframalpha.com/input/?i=rref+%7B%282%2C1%2C5%29%2C+%28-1%2C-1%2C10%29%2C+%288%2C5%2C-5%29%7D
Sorry if I sound persistent, but I still don't understand (I have looked through the link).
  • phi
what kind of math are you studying?
Vectors, of Vectors and Calculus.
Specifically, Lines and Planes.
  • phi
Do they teach you about independent vectors? How to tell if two vectors are independent?
I had difficulties understanding the course. I don't exactly know.
equations of lines is the lesson (specifically)
  • phi
or maybe more simple: find the equation of a line through 2 of the points, and show the 3rd point does not satisfy the equation. So you know all 3 points do not lie on the same line, and therefore define a plane.
What's an equation of a line?
I'm just looking thru my textbook here, but I can only see equation of a plane. I'll use Google now though.
y=mx+b? How does that work for a 3-space point?
  • phi
I think you do A + n(B-A) e.g. (2,1,5)+n ((-1,-1,10)-(2,1,5)) where n is any value (a scalar)
You mean like a vector equation? [x,y,z]=[xo,y0,zo]+t[a1,a2,a3]+s[b1,b2,b3]
But that wouldn't make sense to me...Aren't the ones with scalar multipliers direction vectors?
  • phi
B-A points in the direction from A to B |dw:1333072072947:dw| you scale it to move along it. Add A so the direction vector starts at A rather than the origin so A+ n(B-A)
  • phi
*should be B-A as the label
Ok. Maybe I should ask what I should actually do for this question then. I don't seem to be able to comprehend our conversation.
I just solve for that? All I had to do was plug them in at random onto a vector equation? Sorry, my eyes are starting to hurt.
  • phi
It looks like the 3 points are not collinear, so they define a plane
Wait, how do I determine collinear? And, do I just do as what I suggested 2 comments above?
  • phi
See if this helps http://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfLines.aspx
How far should I read into this?
  • phi
Start at the paragraph below the ellipse.
Ok. I just figured out that the equation of the line is the same as the vector equation.
  • phi
Here is the equation of the line through points (2,1,5) and (-1,-1,10) (2,1,5) + n( -1 -2, -1-1, 10-5) (2,1,5) +n( -3, -2, 5) Is there an n that gets us to the point (8,5,-5)? n= -2 gives us (2,1,5)-2(-3,-2,5) = (2+6, 1+4,5-10)= (8,5,-5) so it looks like there are all on the same line.
( -1 -2, -1-1, 10-5) I'm confused about that. I was never taught to do that in class. Also, "Is there an n that gets us to the point (8,5,-5)? n= -2 gives us (2,1,5)-2(-3,-2,5) = (2+6, 1+4,5-10)= (8,5,-5)" Do I figure out n's value by guess and check?
  • phi
No you don't guess. you have 3 separate equations for x,y,z example 2+n(-3) = 8 is the first one. It requires n= -2. it turns out all of the equations work for -2. so point (8,5,-5) is on the line
  • phi
that is point (-1,-1,10) - (2,1,5) rewritten as ( -1 -2, -1-1, 10-5)
  • phi
Look at Example 1 in Paul's notes
I don't even understand that...
IS he adding the 2 points or subtracting them?
  • phi
subtracting the 2 points. Maybe 2-d is easier? say you are at (1,1) and you want to get to (2,2) how much to you move in the x and y? subtract to find you have to move (1,1) the (1,1) represents your slope. to get from one point on the line to another move over 1 and up 1. Or scale the 1,1 to get to any point on the line. example you are at 1,1 and you move 1/2 over and 1/2 up. You are still on the line
Ok. That makes sense I guess. I hope.
  • phi
To continue the 2-d example. if we say (x,y)= n(1,1) this means all points where y=x Say we want the line y= x+1 then we could say (x,y)= (0,1)+n(1,1)
  • phi
Good luck. At least you have the answer, It is not a plane.
Ok. Thanks for the help, and putting up with my lack of knowledge on the topic.
  • phi
Does this help?
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