Find a formula for the transformation that rotates -60 degrees about 1-3i
To find a formula for a rotation, we can use the concept of complex numbers and the multiplication of complex numbers.
A rotation of -60 degrees about the origin in the complex plane corresponds to multiplying a complex number by a unit vector representing the angle of rotation. In our case, we want to rotate -60 degrees about the point 1-3i, so we need to translate the rotation center to the origin, perform the rotation, and then translate back to the original position.
Let's break down the steps:
1. Translate the rotation center to the origin:
To move the point 1-3i to the origin, we subtract it from the complex number we want to rotate. Let's call the number we want to rotate "z."
So, the translated complex number will be z' = z - (1-3i)
2. Perform a rotation of -60 degrees about the origin:
To rotate a complex number by an angle θ, we can multiply it by a unit vector e^(iθ), where i is the imaginary unit and θ is the angle of rotation in radians.
In our case, we want to rotate -60 degrees, which is equivalent to -π/3 radians.
So, the rotated complex number will be z'' = z' * e^(i(-π/3))
3. Translate back to the original position:
Finally, to move the rotated complex number back to the original position, we add the rotation center that we subtracted in step 1.
The formula for the transformation that rotates -60 degrees about 1-3i is:
z'' + (1-3i)
Simplifying the above steps, we have:
z'' = (z - (1-3i)) * e^(i(-π/3))
So, the formula for the transformation that rotates -60 degrees about 1-3i is:
z'' = (z - (1-3i)) * e^(i(-π/3)) + (1-3i)
You can substitute any complex number z into this formula to compute its rotated position.