SOLUTION: I have this problem for homework and I have utterly no idea how to go about this. My tutor (who assigned it in the first place) even told me to guess on the answers. Even still, I

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: I have this problem for homework and I have utterly no idea how to go about this. My tutor (who assigned it in the first place) even told me to guess on the answers. Even still, I       Log On


   

Question 184328: I have this problem for homework and I have utterly no idea how to go about this. My tutor (who assigned it in the first place) even told me to guess on the answers. Even still, I don't have a clue what it means. It's two parts:
a) Suppose that a+bi is multiplied by 1+i. By how many degrees must the arrow from (0,0) to (a,b) be rotated to coincide with the arrow from (0,0) to the product?
b) Suppose that a+bi is multiplied by √3+i. By how many degrees must the arrow from (0,0) to (a,b) be rotated to coincide with the arrow from (0,0) to the product?
To start, I thought the most rational thing to do was to multiply (for part a) a+bi by 1+i, and that gets me a+ai+bi-b. But I don't know what to do after that! I guess if I understood part a, I would be able to do part b, but I thought I should write the whole thing just in case.
Please help!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first part to get you started.


First, multiply a%2Bbi and 1%2Bi to get:


%28a%2Bbi%29%281%2Bi%29=a%2Bai%2Bbi%2Bbi%5E2=a%2Bai%2Bbi-b (which is what you have)



%28a%2Bbi%29%281%2Bi%29=%28a-b%29%2B%28a%2Bb%29i Rearrange and group like terms


So we essentially have two complex numbers a%2Bbi and %28a-b%29%2B%28a%2Bb%29i


If we plot these numbers on the complex plane, then we'll have the points (a,b) and (a-b,a+b)


Now think of these points as vectors


So we have two vectors and


So the goal now is to find the angle between the two vectors (so we'll know the exact angle needed to rotate)


Remember, the angle between two vectors can be found through the formula

where is the magnitude (ie length) of vector and is the magnitude (ie length) of vector


Note: I'm making assumptions that you are familiar with vector calculations (such as the dot product and finding the magnitude of a vector). If my assumptions are wrong, please let me know.


So let's calculate the dot product:







Now let's calculate the magnitudes:






Now multiply the magnitudes




Now plug these values into the given formula above




Cancel out the common terms






Now take the arccosine of both sides (to solve for theta)




Evaluate the arccosine of 1%2Fsqrt%282%29 to get pi%2F4 radians (or 45 degrees). Note: use the unit circle

or



=====================================================

Answer:

So the vector needs to rotate pi%2F4 radians (or 45 degrees) in order to be parallel to the vector



Now if any of this is not making any sense, simply pick values for "a" and "b" and plug them in. So let's say that we choose a=2 and b=3. So the first number would be 2%2B3i and the second number would be %282-3%29%2B%282%2B3%29i=-1%2B5i

Now find the angle between the vectors <2,3> and <-1,5> and you'll find that it will be 45 degrees.