Question

Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) are collinear.

Answer 1

The given points are A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1).

∴ `vec(AB) = (2 - 1)hati + (6 - 2)hatj + (3 - 7)hatk = hati + 4hatj - 4hatk`

`vec(BC) = (3 - 2)hati + (10 - 6)hatj + (-1 - 3)hatk = hati + 4hatj - 4hatk`

`vec(AC) = (3 - 1)hati + (10 - 2)hatj + (-1 - 7)hatk = 2hati + 8hatj - 8hatk`

`|vec(AB)| = sqrt(1^2 + 4^2 + (-4)^2) `

`= sqrt(1 + 16 + 16) `

`= sqrt33`

`|vec(BC)| = sqrt(1^2 + 4^2 + (-4)^2)`

` = sqrt(1 + 16 + 16) `

`= sqrt33`

`|vec(AC)| = sqrt(2^2 + 8^2 + 8^2)`

` = sqrt(4 + 64 + 64) = sqrt(132) `

`= 2sqrt33`

∴ `|vec(AC)| = |vec(AB)| + vec(BC)`

Hence, the given points A, B and C are collinear.