Show that the points A(2, 1, 3), B(5, 0, 5) and C(-4, 3, -1) are collinear.
Given -
A = (2,1,3)
B = (5,0,5)
C = (-4,3,-1)
To prove – A, B and C are collinear
Formula to be used – If P = (a,b,c) and Q = (a’,b’,c’),then the direction ratios of the line PQ is given by ((a’-a),(b’-b),(c’-c))
The direction ratios of the line AB can be given by
((5-2),(0-1),(5-3))
=(3,-1,-2)
Similarly, the direction ratios of the line BC can be given by
((-4-5),(3-0),(-1-5))
=(-9,3,-6)
Tip – If it is shown that direction ratios of AB=λ times that of BC , where λ is any arbitrary constant, then the condition is sufficient to conclude that points A, B and C will be collinear.
So, d.r. of AB
=(3,-1,-2)
=(-1/3)Χ(-9,3,-6)
=(-1/3)Хd.r. of BC
Hence, A, B and C are collinear