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Collinear Vectors

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Question

Let $\to a, \to b, \to c$ are three non zero vectors such that any two of them are non-collinear. If $\to a + \to b$ is collinear with $\to c$ and $\to b + \to c$ is collinear with $\to a$ , then the value of $\to a + \to b + \to c$ equals

A

3 \to a

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B

3 \to b

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C

3 \to c

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D

0

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Solution

The correct option is D $0$
We have, $\to a + \to b$ is collinear with $\to c$

Then, $\to a + \to b = u \to c . . . . . . (1)$

And $\to b + \to c$ is collinear with $\to a$

Then, $\to b + \to c = v \to a . . . . . . (2)$

By equation (1) and (2) to, we get

$\to b + \to c = v (u \to c - \to b)$

$\to b + \to c = v u \to c - v \to b$

$\to b + v \to b = v u \to c - \to c$

$\to b (1 + v) = \to c (u v - 1)$

But given that, $\to b$ and $\to c$ are collinear.

Then, $\to b = 0$ and $\to c = 0$

Now, $1 + v = 0, v = - 1$

And $u v - 1 = 0, u \times (- 1) = 1, u = - 1$

Put the value of u and v in equation (1) and (2), we get

$\to a + \to b = (- 1) \to c$

$\to a + \to b = - \to c$

$\to a + \to b + \to c = 0$
Hence, this is the answer.

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