Question

If the vertices of a square z1,z2,z3 and z4 taken in the anti-clockwise order, then z3=

• A. -iz1-(1+i)z2
• B. z1-(1+i)z2
• C. z1+(1+i)z2
• D. -iz1+(1+i)z2

Answer 1

Answer: (D)

-iz1+(1+i)z2

Answer 2

The expression z3 = -iz1 + (1+i)z2 represents the location of vertex z3 of a square when the vertices are labeled in anti-clockwise order.

-iz1: This term represents a rotation of z1 by 90 degrees in the counterclockwise direction around the origin. Since the vertices are listed in anti-clockwise order, moving from z1 to z3 involves a 90-degree rotation from z1.

(1+i)z2: This term represents a translation of z2 by a vector (1+i), which corresponds to moving one unit in the positive real direction and one unit in the positive imaginary direction from z2. This translation from z2 also contributes to reaching the position of z3.

Combining the effects of the 90-degree rotation (-iz1) and the translation by (1+i)z2, we arrive at the correct position for z3 when the vertices are listed in anti-clockwise order for a square.

This answer satisfies the requirement that z3 should be diagonally opposite to z1, and it also accounts for the position of z2 in the anti-clockwise order of vertices.

The correct answer is option (D): -iz1+(1+i)z2