First slide

Algebra of vectors

Question

Let ABCD be a quadrilateral. Let E be the point of intersection of the line joining the midpoints of the opposite sides. If O is any point and $\vec{OA} + \vec{OB} + \vec{OC} + \vec{OD} = k \vec{OE}$ then is k equal to

Moderate

A

3
B

9
C

7
D

4

Solution

Let $\vec{OA} = \vec{a}, \vec{OB} = \vec{b}, \vec{OC} = \vec{c} and \vec{OD} = \vec{d}$
Therefore, $\vec{OA} + \vec{OB} + \vec{OC} + \vec{OD} = \vec{a} + \vec{b} + \vec{c} + \vec{d}$
P, the midpoint of AB ,is $\frac{\vec{a} + \vec{b}}{2}$
Q, the midpoint of $\vec{C D}, is \frac{\vec{c} + \vec{d}}{2}$
Therefore, the midpoint of $\vec{PQ} is \frac{\vec{a} + \vec{b} + \vec{c} + \vec{d}}{4}$
Similarly, the midpoint of RS is $\frac{\vec{a} + \vec{b} + \vec{c} + \vec{d}}{4}$
$i.e., \vec{OE} = \frac{\vec{a} + \vec{b} + \vec{c} + \vec{d}}{4} \Rightarrow k = 4 .$

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