I was working on a question and am confused , the question is: P6: 3 a) A collec

Question

I was working on a question and am confused , the question is:

P6: 3 a) A collection of points is collinear if they sit on a common line. As we have seen, two points in R" are always on a common line. What about three pointsin R", are they always on a common line? Explain why, or give a counter-example for the smallest possible n. 2 b) A collection of points is coplanar if they sit on a common plane (2-plane). Verify three points in R" always sit on a common 2-plane. Do four points always sit on a common 2-plane? Explain why, or give a counter-example for the smallest possible n

Explanation / Answer

a) Three points in Rn need not be in the same line for n>=2

For n=1 ,sa it is one dimentional all points will be on sane line.So,

n=2 is the smallest possible value of n to show counter example

counter example:

consider XY plane ,consider the points (1,0) ,(-1,0) ,(0,1) these points don't lie on same line

b)Yes three points on Rn always lie on the the same plane

proof:

Join the three points we will get a triangle extend it in all directions parallel to triangle , you will a plane.

No, we can't say that four points lie on a common 2-plane.

counter example:

n=3 is smallest value to show counter example

consider XYZ coordinate system,

consider the points (1,0,0) ,(-1,0,0) ,(0,1,0) ,(0,0,1) will not lie on same plane.

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