In the argand plane, a vector OA, where O represents the origin and A represents the complex num

step 1 Number 59 here in this problem we have to choose the correct option and we are given in the Argo

In the argand plane, a vector OA, where O represents the...

In the argand plane, a vector $\overline{\mathrm{OA}}$, where $\mathrm{O}$ represents the origin and $\mathrm{A}$ represents the complex number $(1+2 \mathrm{i})$ is turned in the clockwise direction through an angle $\frac{\pi}{4}$ and then stretched $\sqrt{2}$ times. The complex number represent- ing the new vector is $\begin{array}{llll}\text { (a) }(3+\mathrm{i}) & \text { (b) }(\sqrt{2}(1+\sqrt{3} \mathrm{i}) & \text { (c) } \sqrt{2}(-2+\mathrm{i}) & \text { (d) } \frac{\sqrt{2}}{2}(1+\sqrt{3} \mathrm{i})\end{array}$

In the argand plane, a vector $\overline{\mathrm{OA}}$, where $\mathrm{O}$ represents the origin and $\mathrm{A}$ represents the complex number $(1+2 \mathrm{i})$ is turned in the clockwise direction through an angle $\frac{\pi}{4}$ and then stretched $\sqrt{2}$ times. The complex number represent-
ing the new vector is
$\begin{array}{llll}\text { (a) }(3+\mathrm{i}) & \text { (b) }(\sqrt{2}(1+\sqrt{3} \mathrm{i}) & \text { (c) } \sqrt{2}(-2+\mathrm{i}) & \text { (d) } \frac{\sqrt{2}}{2}(1+\sqrt{3} \mathrm{i})\end{array}$

Key Concepts

Complex Multiplication as a Combination of Rotation and Scaling

Complex multiplication combines rotation and scaling in a single operation. When two complex numbers are multiplied, their moduli (magnitudes) are multiplied and their arguments (angles) are added. This intrinsic relationship between polar forms and multiplication makes it a powerful tool for performing geometric transformations in the plane.

Scaling (Dilation) in the Complex Plane

Scaling, or dilation, in the context of complex numbers is the transformation that changes the distance of a point from the origin by a constant factor. This is accomplished by multiplying the complex number by a positive real number, which stretches or compresses the vector without affecting its direction.

Rotation in the Complex Plane

Rotation in the complex plane involves turning a complex number about the origin by a given angle. This rotation is achieved by multiplying the complex number by another complex number of unit modulus whose argument is equal to the rotation angle. This operation shifts the angle while preserving the distance from the origin (modulus), representing a pure rotation.

Argand Plane

The Argand plane is a geometric representation of complex numbers where each number is represented as a point in the plane. The horizontal axis corresponds to the real part and the vertical axis corresponds to the imaginary part of the complex number, allowing one to visualize operations on complex numbers as geometric transformations.

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