BodhiAI

Nov. 11, 2019
**Rotation theorem in complex numbers**

Let AB and CD represent two lines on the Argand Plane. The vertices A, B, C and D are represented by affixes and^{ z4 }respectively.

Such that the lines AB and CD are inclined at an angle to each other. Let us transform the lines in such a way to a new position, then vertices A and C lie on origin such that the other ends have affixes and respectively by vertices M and N respectively in new position. Now, the line ON is rotated in such a way in the anticlockwise direction by an angle so that it lies along ON.

Let θ_{1 }and θ_{2} be the argument of and respectively,

then,

arg

Problem Solving Trick

If is the angle between lines passing through A, B and C, D where and z_{4}_{ are }the affixes of points A, B, C and D respectively, then

and the equivalent complex number

where, α is the angle with which the line AB rotated in the anticlockwise direction lies along CD. Now, if the line AB is rotated in the clockwise direction by an angle then also it can lie along CD, then

and the equivalent complex number is