Question

If a, b and c are three non-zero vectors which are pairwise non-collinear. If a + 3b is collinear with c and b + 2c is collinear with a, then a + 3b + 6c is ______.

Options

• c

• 0

• a + c

• a

Answer 1

If a, b and c are three non-zero vectors which are pairwise non-collinear. If a + 3b is collinear with c and b + 2c is collinear with a, then a + 3b + 6c is 0.

Explanation:

Given, a + 3b is collinear with c.

So, a + 3b = λc  ...(i)

where λ is some real number.

Similarly, b + 2c = µa   ...(ii)

Adding equations, (i) and (ii), we get

a + 4b + 2c = λc + µa

`\implies` (1 – µ) a + 4b + (2 – λ)c = 0

Multiply by `3/4`, we get

`3/1(1 - µ)a + 3b + 3/4 (2 - λ)c` = 0

∴ `3/4(1 - µ) = 1` and `3/4 (2 - λ)` = 6

`\implies` 3 – 3µ = 4 and 2 – λ = 8

`\implies` µ = `(-1)/3` and λ = – 6 

`\implies` a + 3b + 6c = 0