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Vector Addition

If C is the...

Question

If $C$ is the mid point of $A B$ and $P$ is any point outside $A B$ , then

A

¯ ¯¯¯¯¯¯ ¯ P A + ¯ ¯¯¯¯¯¯ ¯ P B + ¯ ¯¯¯¯¯¯ ¯ P C = 0

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B

¯ ¯¯¯¯¯¯ ¯ P A + ¯ ¯¯¯¯¯¯ ¯ P B + 2 ¯ ¯¯¯¯¯¯ ¯ P C = ¯ 0

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C

¯ ¯¯¯¯¯¯ ¯ P A + ¯ ¯¯¯¯¯¯ ¯ P B = ¯ ¯¯¯¯¯¯ ¯ P C

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D

¯ ¯¯¯¯¯¯ ¯ P A + ¯ ¯¯¯¯¯¯ ¯ P B = 2 ¯ ¯¯¯¯¯¯ ¯ P C

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Solution

The correct option is D $¯ ¯¯¯¯¯¯ ¯ P A + ¯ ¯¯¯¯¯¯ ¯ P B = 2 ¯ ¯¯¯¯¯¯ ¯ P C$
Let $\to p, \to a, \to b, \to c$ be the position vectors of points $P, A, B, C$ respectively.
$L . H . S = ¯ ¯¯¯¯¯¯ ¯ P A + ¯ ¯¯¯¯¯¯ ¯ P B = \to p - \to a + \to p - \to b = 2 (\to p - \frac{\to a + \to b}{2}) = 2 \to P C$

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