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Question

Let $\to a, \to b$ and $\to c$ be three non zero vectors which are pairwise noncollinear. If $\to a + 3 \to b$ is collinear with $\to c$ and $\to b + 2 \to c$ is collinear with $\to a$ , then $\to a + 3 \to b + 6 \to c$ is:

A

\to a + \to c

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B

\to a

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C

\to c

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D

0

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Solution

The correct option is D $0$
As $\to a + 3 \to b$ is collinear with $\to c$ ;
$\to a + 3 \to b = λ \to c \dots (i)$
As $\to b + 2 \to c$ is collinear with $\to a$ ;
$\to b + 2 \to c = μ \to a \dots (i i)$
From $(i)$ we have $\to a + 3 \to b + 6 \to c = (λ + 6) \to c \dots (i i i)$
and from $(i i)$ we have $\to a + 3 \to b + 6 \to c = (1 + 3 μ) \to a \dots (i v)$
Since $\to a$ is non collinear with $\to c$ we have $λ + 6 = 1 + 3 μ = 0$
hence from $(i i i)$ or $(i v)$ we have $\to a + 3 \to b + 6 \to c = 0$

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