In each of the following problems, show by vector method that the given points are collinear. A(2,6,3), B(1,2,7) and C(3,10,-1)

Explanation

Given that A = (2,6,3), B = (1,2,7) and C = (3,10,-1)
Then `vec(AB) = (1-2)hati+(2-6)hatj+(7-3)hatk = -hati-4hatj+4hatk`
`vec(BC) = (3-1)hati=(10-2)hatj+(-1-7)hatk = 2hati+8hatj-8hatk`
= `-2(-hati-4hatj+4hatk) = -2vec(AB)`
As `vec(BC) is a scalar multiple of `vec(AB),` it is parallel to `vec(AB).
Again `vec(AB)` and `vec(BC)` have a common point.
So `vec(AB)` is along `vec(BC)`.
Hence the points A, B, C are collinear. (Proved).