If the points A (3, 0, p), B (- 1, q, 3) and C (- 3, 3, 0) are collinear, then find

(i) the ratio in which the point C divides the line segment AB

(ii) the values of p and q.

#### Solution

Let `bar"a", bar"b", bar"c"` be the position vectors of A, B and C respectively.

Then `bar"a" = 3hat"i" + 0.hat"j" + "p"hat"k"`,

`bar"b" = - hat"i" + "q"hat"j" + 3hat"k"` and

`bar"c" = -3hat"i" + 3hat"j" + 0.hat"k"`

**(i) **As the points A, B, C are collinear, suppose the point C divides line segment AB in the ratio λ : 1.

∴ by the section formula,

`bar"c" = (lambda.bar"b" + 1.bar"a")/(lambda + 1)`

∴ `- 3hat"i" + 3hat"j" + 0.hat"k"`

`= (lambda(- hat"i" + "q"hat"j" + 3hat"k") + (3hat"i" + 0.hat"j" + "p"hat"k"))/(lambda + 1)`

∴ `(lambda + 1)(- 3hat"i" + 3hat"j" + 0.hat"k") = (-lambdahat"i" + lambda"q"hat"j" + 3lambdahat"k") + (3hat"i" + 0.hat"j" + "p"hat"k")`

∴ `-3 (lambda + 1)hat"i" + 3(lambda + 1)hat"j" + 0.hat"k" = (- lambda + 3)hat"i" + lambda"q"hat"j" + (3lambda + "p")hat"k"`

By equality of vectors, we have,

- 3(λ + 1) = - λ + 3 ....(1)

3(λ + 1) = λq .....(2)

0 = 3λ + p ....(3)

From equation (1), - 3λ - 3 = - λ + 3

∴ - 2λ = 6

∴ λ = - 3

∴ C divides segment AB externally in the ratio 3 : 1.

**(ii) **Putting λ = - 3 in equation (2), we get

3(- 3 + 1) = -3q

∴ - 6 = - 3q

∴ q = 2

Also, putting λ = - 3 in equation (3), we get

0 = - 9 + p

∴ p = 9

Hence p = 9 and q = 2.