JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Complex Numbers

Let

w_{1}

be the point obtained by the rotation of

z_{1} = 5 + 4 i

about the origin through a right angle in the anticlockwise direction, and

w_{2}

be the point obtained by the rotation of

z_{2} = 3 + 5 i

about the origin through a right angle in the clockwise direction. Then the principal argument of

w_{1} - w_{2}

is equal to

Answer: (D)

Solution

Rotation by

9 0^{\circ}

anticlockwise is multiplication by

i

w_{1} = z_{1} \cdot i = (5 + 4 i) i = - 4 + 5 i

. Rotation by

9 0^{\circ}

clockwise is multiplication by

- i

w_{2} = z_{2} \cdot (- i) = (3 + 5 i) (- i) = 5 - 3 i

. Then

w_{1} - w_{2} = (- 4 + 5 i) - (5 - 3 i) = - 9 + 8 i

. This lies in the second quadrant. Principal argument =

π - tan^{- 1} (\frac{8}{9})

JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Single correct option