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Vector Addition

z1, z2, z3, z...

Question

$z_{1}, z_{2}, z_{3}, z_{4}$ are distinct complex numbers representing the vertices of a quadrilateral $A B C D$ taken in order. If $z_{1} - z_{4} = z_{2} - z_{3}$ and $a r g [\frac{(z_{4} - z_{1})}{(z_{2} - z_{1})}] = π / 2$ , then the quadrilateral is

A

rectangle

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B

rhombus

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C

square

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D

trapezium

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Solution

The correct option is B rectangle
Let
$A = z_{1}$
$B = z_{2}$
$C = z_{3}$
$D = z_{4}$
Hence
$z_{1} - z_{4} = z_{2} - z_{3}$
Implies $A D | | B C$
Hence opposites sides are parallel. Therefore its a type of parallelogram.
Now
$a r g (\frac{z_{4} - z_{1}}{z_{2} - z_{1}}) = ∠ A$
$= \frac{π}{2}$ .
Hence
All the angles are $90^{0}$
Therefore, it is rectangle.

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