If the points with position vectors and are collinear, find the value of a.

Let us understand that, two more points are said to be collinear if they all lie on a single straight line.

Let the points be A, B and C having position vectors such that,

So, let us find and .

Therefore, is given by

…(i)

And is given by

…(ii)

Since, it has been given that points A, B and C are collinear.

So, we can write as

Where λ = a scalar quantity

Put the values of and from (i) and (ii), we get

Comparing the vectors and respectively, we get

a – 12 = 2λ …(iii)

and, 16 = –8λ

From –8λ = 16, we can find the value of λ.

–8λ = 16

⇒ λ = –2

Put λ = –2 in equation (iii), we get

a – 12 = 2λ

⇒ a – 12 = 2(–2)

⇒ a – 12 = –4

⇒ a = –4 + 12

⇒ a = 8

Thus, we have got a = 8.