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Addition of Vectors

If A,B,C,D ...

Question

If $A, B, C, D$ be any four points and $E$ and $F$ be the mid-points of $A C$ and $B D$ , respectively, then $\to A B + \to C B + \to C D + \to A D$ is equal to

A

3 \to E F

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B

4 \to E F

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C

4 \to F E

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D

3 \to F E

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Solution

The correct option is B $4 \to E F$
Solution:- (C) $4 \to E F$
$A B C D$ is a quadrilateral.
Given that $F$ is the mid-point of $B D$ , we get
$\to A B + \to A D = 2 \to A F . . . . . (1)$
Similarly $\to C B + \to C D = 2 \to C F . . . . . (2) [E is the mid point of AC]$
Adding ${e q}^{n} (1) & (2)$ , we have
$\to A B + \to A D + \to C B + \to C D = 2 (\to A F + \to C F)$
$\to A B + \to A D + \to C B + \to C D = 2 ((- \to F A) + (- \to F C))$
$\to A B + \to A D + \to C B + \to C D = - 2 (\to F A + \to F C)$
$\to A B + \to A D + \to C B + \to C D = - 2 (2 \to F E)$
$\to A B + \to A D + \to C B + \to C D = - 2 (- 2 \to E F)$
$\to A B + \to A D + \to C B + \to C D = 4 \to E F$

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