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Test for Collinearity of Vectors

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Question

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

A

λ \to a

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B

λ \to b

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C

λ \to c

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D

0

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Solution

The correct option is D $0$
If $\to a + 2 \to b$ is collinear with $\to c$ , then $\to a + 2 \to b = t \to c$ ..... $. (1)$
Also if $\to b + 3 \to c$ is collinear with $\to a$ , then $\to b + 3 \to c = λ \to a$
$\Rightarrow \to b = λ \to a - 3 \to c$
On putting this value in equation $(1)$
$\to a + 2 (λ \to a - 3 \to c) = t \to c \Rightarrow \to a + 2 λ \to a - 6 \to c = t \to c \Rightarrow (\to a - 6 \to c) = t \to c - 2 λ \to a$
On comparing, we get
$1 = 2 λ$
$\Rightarrow λ = - 1 / 2$
and $- 6 = t \Rightarrow t = - 6$
From $(1)$
$\to a + 2 \to b = - 6 \to c$
$\Rightarrow \to a + 2 \to b + 6 \to c = 0$

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