how do you find if any given 3 points (in 3D) lie on a straight line

Question

anonymous · Answer

...? np just doing some calc problems, I'm given 3 points and need to find if all three lie on a straight line

anonymous · Answer

is it by finding midpoints?

anonymous · Answer

Okay, so if you find the difference between two points... that will give you a vector.

anonymous · Answer

So get two vectors from the three points.

anonymous · Answer

If you do the cross product and get 0 then they're parallel

anonymous · Answer

Parallel vectors which go through the same point are on the same line.

anonymous · Answer

so out of the 3 points, use 2 to find a vector, then us another set of 2 from the 3 not used in the first one to find another vector and if the quell then all 3 points are in a line?

anonymous · Answer

No, from the two points get two vectors.  Use the cross product of the two vectors to find out if they're parallel

anonymous · Answer

but i got 3 points, so pick any 2?

anonymous · Answer

Pick any two for the first vector, and then change one of the points for the second vector.

anonymous · Answer

Okay suppose your points are $P_1,P_2, P_3$.
Let:$$
\mathbf v_1=P_2-P_1\
\mathbf v_2=P_3-P_1
$$If: $$
\mathbf v_1\times \mathbf v_2=\mathbf 0
$$Then $P_1,P_2, P_3$ are on the same line.

anonymous · Answer

i got the cross product, and now i square all 3 and square root that to see if it = 0 right?