Let ABCD be a quadrilateral. If E and F are the mid points of the diagonals AC and BD respectively and $(\overrightarrow{\mathrm{AB}}-\overrightarrow{\mathrm{BC}})+(\overrightarrow{\mathrm{AD}}-\overrightarrow{\mathrm{DC}})=\mathrm{k} \quad \overrightarrow{\mathrm{FE}}$, then k is equal to

JEE Main · 2023 · 15 Apr 2023 · Shift 1 · Vector Algebra

Let ABCD be a quadrilateral. If E and F are the mid points of the diagonals AC and BD respectively and

(AB - BC) + (AD - DC) = k FE

, then k is equal to

Answer: (C)

Express the vectors through position vectors

a, b, c, d

. Then

A B - B C = (b - a) - (c - b) = 2 b - a - c

and

A D - D C = (d - a) - (c - d) = 2 d - a - c

. Adding, the left side is

2 (b + d) - 2 (a + c)

. Since

E

is the midpoint of

A C

and

F

the midpoint of

B D

e = \frac{a + c}{2}

and

f = \frac{b + d}{2}

, so the left side equals

4 f - 4 e = - 4 (e - f) = - 4 F E

. Comparing with

k F E

gives

k = - 4

JEE Main · 2023 · 15 Apr 2023 · Shift 1 · Single correct option