Question

Determine whether the points lie on a straight line. (a) a(2, 5, 3), b(3, 6, 2), c(1, 4, 4)

Answer 1

All three points lie on a straight line.  
The equation for a line is:
 (x-x0)/L = (y-y0)/M = (z-z0)/N
 where
 (L,M,N) = the direction vector for that line.
 (x,y,z) = any point on the line.
 (x0,y0,z0) = a known point on the line.
 
 So let's create the equation for a line passing through points a and b and see
if point c lies on that line.
 (L,M,N) = a - b
 (L,M,N) = (2, 5, 3) - (3, 6, 2) = (2 - 3, 5 - 6, 3 - 2) = (-1, -1, 1)
 So we have
 (x-2)/-1 = (y-5)/-1 = (z - 3)/1
 -(x-2) = -(y-5) = (z - 3)
 
 Let's check the points.
 a(2,5,3)
 -(x-2) = -(y-5) = (z - 3)
 -(2-2) = -(5-5) = (3 - 3)
 0 = 0 = 0
 The above equation is true, so point a lies on the line.
 
 b(3,6,2)
 -(x-2) = -(y-5) = (z - 3)
 -(3-2) = -(6-5) = (2 - 3)
 -1 = -1 = -1
 The above equation is true, so point b lies on the line.
 
 c(1,4,4)
 -(x-2) = -(y-5) = (z - 3)
 -(1-2) = -(4-5) = (4 - 3)
 1 = 1 = 1
 The above equation is true, so point c lies on the line.
 
 All three points create true expressions for the formula, so all three points lie
on a straight line.