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Addition of Vectors

ABCD is a qua...

Question

$A B C D$ is a quadrilateral. $E$ is the point of intersection of the line joining the midpoint of the opposite sides. If $O$ is any point and $- - \to O A + - - \to O B + - - \to O C + - - \to O D = x - - \to O E$ , then $x$ is equal to

3

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4

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Solution

The correct option is D $4$
Let
$- - \to O A = \to a, - - \to O B = \to b, - - \to O C = \to c$ and $- - \to O D = \to d$

Therefore,
$- - \to O A + - - \to O B + - - \to O C + - - \to O D = \to a + \to b + \to c + \to d$
Let $P$ , is the midpoint of $A B$ ,
$= \frac{\to a + \to b}{2}$
Let $Q$ , is the midpoint of $C D$ ,
$= \frac{\to c + \to d}{2}$
Therefore, the midpoint of $- - \to P Q$ is $\frac{\to a + \to b + \to c + \to d}{4}$

Similarly, the midpoint of $- - \to R S$ is $\frac{\to a + \to b + \to c + \to d}{4}$
$\Rightarrow - - \to O E = \frac{\to a + \to b + \to c + \to d}{4}$
$\Rightarrow x = 4$

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