Question

Let vector(a, b, c) be three non-zero vectors such that any two of them are non-collinear. If vector(a + 2b)  is collinear with vector c and vector(b + 3c)  is collinear with vector a, then prove that vector(a + 2b + 6c) = 0.

Answer 1

It is given that vector(a + 2b)  is collinear with vector c, so

vector a + 2b = λc (for some scalar λ)   (1)

Also vector b + 3c is collinear with a, so

vector b + 3c = μa (for some scalar μ)   (2)

From Eqs. (1) and (2), we get

(1 + 2μ)b + (3 - μλ)c = 0

⇒ 1 + 2μ = 0 and 3 - μλ = 0 {b and c are non-collinear vectors}

⇒  μ = - 1/2 and λ = - 6

Substituting the values of λ and μ in Eqs. (1) and (2), we get

a + 2b + 6c = 0