Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero vectors such that no two of these are collinear. If the vector $\vec{a}+2 \vec{b}$ is collinear with $\vec{c}$ and $\vec{b}+3 \vec{c}$ is collinear with $\vec{a}$ ( $\lambda$ being some non-zero scalar) then $\vec{a}+2 \vec{b}+6 \vec{c}$ equals

(B) $\lambda \overrightarrow{\mathrm{b}}$

(C) $\lambda \overrightarrow{\mathrm{c}}$

JEE Main · 2004 · 24 Apr 2004 · Offline · Vector Algebra

Let

a, b

and

c

be three non-zero vectors such that no two of these are collinear. If the vector

a + 2 b

is collinear with

c

and

b + 3 c

is collinear with

a

(

λ

being some non-zero scalar) then

a + 2 b + 6 c

equals

Answer: (D)

Collinearity of

a + 2 b

with

c

lets us write

a + 2 b = μ c

, and collinearity of

b + 3 c

with

a

gives

b + 3 c = ν a

, so

b = ν a - 3 c

. Substituting,

a + 2 (ν a - 3 c) = μ c

becomes

(1 + 2 ν) a = (μ + 6) c

. As

a

and

c

are not collinear, both sides must vanish, so

1 + 2 ν = 0

and

μ + 6 = 0

, giving

μ = - 6

. Then

a + 2 b = μ c = - 6 c

, which means

a + 2 b + 6 c = 0

JEE Main · 2004 · 24 Apr 2004 · Offline · Single correct option