# Rotation of complex number

• AlexChan
In summary, given A(2√3,1) in R^2 , rotate OA by 30° in clockwise direction and stretch the resulting vector by a factor of 6 to OB. The coordinates of B can be expressed in surd form using complex number technique.

#### AlexChan

Given A(2√3,1) in R^2 , rotate OA by 30° in clockwise direction and stretch the resulting vector by a factor of 6 to OB. Determine the coordinates of B in surd form using complex number technique.

i try to rewrite in Euler's form and I found the modulus was √13 but the argument could not be represented in radian, so i feel confuse to cope with this question.

MENTOR Note: moved here from Linear Algebra hence no template

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Did you try drawing it to see where you went wrong?

jedishrfu said:
Did you try drawing it to see where you went wrong?

yes, i draw the it in 2d plane with real and img. axis and roughly get the resulting vector, but still confuse with getting the exact number of the argument

Can you show us what you did step by step?

jedishrfu said:
Can you show us what you did step by step?

the above is the roughly work. I found the angle(argument) between OA and x-axis (real axis) was tan^-1(1/2√3)≈16.10211375
After rotate 30° clockwise, the angle between OB and x-axis was 16.10211375-30≈-13.89788625
and here i got trouble with the representation in surd form.
OA = 2√3+i =√13(cos16.10211375+isin16.10211375)
I found that complex number multiplication represent scaling and rotation, but my notes didnt contain explanation of this part

So the rotated number is sqrt(13)*(cos(-13.89) + I sin(-13.89) ) right?

Then take the negative out of the sin and cos to get it into better form.

Next you have to make it 6 times bigger.

AlexChan said:
Given A(2√3,1) in R^2 , rotate OA by 30° in clockwise direction and stretch the resulting vector by a factor of 6 to OB. Determine the coordinates of B in surd form using complex number technique.

i try to rewrite in Euler's form and I found the modulus was √13 but the argument could not be represented in radian, so i feel confuse to cope with this question.

MENTOR Note: moved here from Linear Algebra hence no template

The argument can be expressed in radians, but you need not bother doing that. Just express the clockwise 30° rotation as multiplication by a complex number of the form a + bi and then carry out the operations of standard complex-number multiplication. Your final answer need not involve any approximate decimal numbers, but can be expressed exactly in terms of square roots and the like.

AlexChan and jedishrfu

## 1. What is rotation in complex numbers?

Rotation in complex numbers refers to the process of changing the direction of a complex number on the complex plane. It involves multiplying the complex number by a rotation matrix, which is determined by the angle of rotation.

## 2. How is rotation of complex numbers represented mathematically?

Rotation of complex numbers is represented mathematically by multiplying the complex number by a rotation matrix. The rotation matrix is a 2x2 matrix that has the cosine and sine of the angle of rotation as its elements.

## 3. What is the relationship between rotation and the modulus of a complex number?

The modulus of a complex number is the distance from the origin to the complex number on the complex plane. When a complex number is rotated, its modulus remains the same, as it only changes direction on the complex plane.

## 4. How does rotation affect the argument of a complex number?

The argument of a complex number is the angle formed between the positive real axis and a line connecting the origin to the complex number on the complex plane. When a complex number is rotated, its argument changes by the same amount as the angle of rotation.

## 5. Can a complex number be rotated by any angle?

Yes, a complex number can be rotated by any angle. This is because the rotation matrix can be adjusted to fit any angle of rotation. Additionally, multiple rotations can be performed on a complex number, resulting in a new complex number with a different rotation and angle.