Show that the points A(2, 3, -4), B(1, -2, 3) and C(3, 8, -11) are collinear.
Show that the points A(2, 3, -4), B(1, -2, 3) and C(3, 8, -11) are collinear.

Answer Given –

A = (2,3,-4)

B = (1,-2,3)

C = (3,8,-11)

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 ((1-2),(-2-3),(3+4))

=(-1,-5,7)

Similarly, the direction ratios of the line BC can be given by ((3-1),(8+2),(-11-3))

=(2,10,-14)

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

=(-1,-5,7)

=(-1/2)Χ(2,10,-14)

=(-1/2)Хd.r. of BC

Hence, A, B and C are collinear