Question

If the points (–1, –1, 2), (2, m, 5) and (3,11, 6) are collinear, find the value of m.

Answer 1

Let the given points be A(–1, –1, 2), B(2, m, 5) and C(3, 11, 6).

Then `vec"AB" = (2 + 1)hat"i" + ("m" + 1)hat"j" + (5 - 2)hat"k"`

= `3hat"i" + ("m" + 1)hat"j" + 3hat"k"`

And `vec"AC" = (3 + 1)hat"i" + (11 + 1)hat"j" + (6 - 2)hat"k"`

= `4hat"i" + 12hat"j" + 4hat"k"`.

Since A, B, C, are collinear

We have `vec"AB" = lambda vec"AC"`

i.e., `(3hat"i" + ("m" + 1)hat"j" + 3hat"k") = lambda(4hat"i" + 12hat"j" + 4hat"k")`

⇒ 3 = `4lambda` and m + 1 = `12lambda`

Therefore m = 8.