Question

Find if three points in space are collinear

Answer 1

Point AA and point BB (A≠BA≠B) determine a line. You can find its equation. See if the coordinates of point C fits the equation. If so, A B and C are colinear, or else, no.

Method 2:

Point AA, BB and CC determine two vectors AB−→−AB→ and AC−→−AC→. Suppose the latter isn't zero vector, see if there is a constant λλ that allows AB−→−=λAC−→−AB→=λAC→.

Other properties if AA, BB and CC are colinear:

∣∣∣∣∣∣AB−→−⋅AC−→−∣∣∣AB−→−∣∣∣⋅∣∣∣AC−→−∣∣∣∣∣∣∣∣∣=1|AB→⋅AC→|AB→|⋅|AC→||=1:\

AB−→−×AC−→−=0→AB→×AC→=0→

Also, two ways to write the equation of a line in 3D:

x−x0a=y−y0b=z−z0cx−x0a=y−y0b=z−z0c

where (x0,y0,z0)(x0,y0,z0) is a point on the line and (a,b,c)(a,b,c) is the direction vector of the line, provided that abc≠0abc≠0.

xyz=x0+at,=y0+bt,=z0+ct.x=x0+at,y=y0+bt,z=z0+ct.

All that remains is calculation.