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Addition of Vectors

Let a⃗,b⃗ a...

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

\to 0

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Solution

The correct option is C $\to 0$
$\to a + 2 \to b = λ \to c - (i)$
$\to b + 3 \to c = μ \to a - (i i)$
$S u b s t i t u t i n g \to a f r o m (i) i n (i i)$
$⟹ \to b + 3 \to c = μ (λ \to c - 2 \to b)$
$⟹ (1 + 2 μ) \to b = (λ μ - 3) \to c$
$⟹ B u t s i n c e \to b a n d \to c a r e n o t c o l l i n e a r .$
$⟹ E i t h e r 1 + 2 μ = 0, λ μ - 3 = 0$
$⟹ μ = \frac{- 1}{2}$
$a n d \frac{- λ}{2} = 3$
$⟹ λ = - 6$

$S u b s t i t u t i n g i n (i)$
$⟹ \to a + 2 \to b + 6 \to c = \to 0$

$∴ D) A n s w e r .$

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