Question

If the points (–1, 2, 3), (4, a, 1) and
(b, 8, 5) are collinear, then a and b are respectively equal to

Answer 1

I can form 2 direction vectors:
directions vector = <4+1, a-2, 1-3) = <5, a-2, -2>
or
direction vector = <b+1, 8-2, 5-3> = <b+1, 6, 2>

so one must be a scalar multiple of the other:
<5, a-2, -2) = k * <b+1, 6, 2>

5 = bk + k
a-2 = -6 -----> a = -4
-2 = 2k ----> k = -1

then
5 = -b - 1 ---> b = -6
a-2 = -6
a = -4

so a = -4, b = -6

check:
Are (-1, 2, 3), (4, -4, 1) and (-6, 8, 5) collinear?
my two direction vectors are:
(5, -6, -2) and (-5, 6, 2) scalar multiples of each other? YES