Question

How will I determine if the given points are collinear or not? For example the given points are A(1, 2, 1), B(4, 7, 0), C(7, 12, -1)..

I know that collinear points must be multiple of the each other....

Answer 1

@SithsAndGiggles can you please help me?

Answer 2

Can you write \(A=kB\), for some scalar \(k\) ? If yes, then they are colinear.

You would do the same to check for \(A\) and \(C\), and again for \(B\) and \(C\).

If you find just one pair to not be colinear, I don't think you'd have to show more work.

Answer 3

for all the coordinates?
can you give another example and show me??

Answer 4

\[A=kB\\
(1,2,1)=k(4,7,0)\]
Matching up components gives you the system
\[\begin{cases}1=4k\\2=7k\\1=0k\end{cases}\]
There is no solution here, so \(A\) and \(B\) are not colinear. Which means none of the points can be colinear.

Answer 5

but it says here in my book it is collinear, I don't know why..

Answer 6

Hey, I only used the definition you gave for colinearity. Do you know if colinearity has anything to do with linear dependence?

Answer 7

I have no idea, I only knew that collinear points must be multiple of the each other and that's it... my book is not discussing linear dependence.

Answer 8

Well, A and B are not colinear according to that description of colinearity. The same work shows that none of the points are colinear.

Answer 9

hmm... I agree to what you said but, I still did not get why are they collinear..

Answer 10

A and B=kA and the origin would be colinear.
If AB does not pass through the origin, then we check the vector P= AB by
P(x,y)=(xb-xa, yb-ya).
Find vector Q=BC similarly.
If A, B and C are colinear, then P=kQ, where \( k \in R \)

Answer 11

can you please explain more... ?

Answer 12

Example:
Let A=(4,6,3), B=(5,7,2), C=(6,8,1)
Then
vector P= vector AB = <5-4, 7-6, 2-3> = <1,1,-1>
vector Q= vector BC = <6-5, 8-7, 1-2> =<1,1,-1>
Since P=kQ where k=1, A,B, C are colinear.

Answer 13

oh ok.. I get it .. thanks..