SOLUTION: Through what positive angle must a vector whose complex expression is -5 - 5i be rotated until it coincides in direction with the vector whose complex expression is 3 + 4i?

Question 1182669: Through what positive angle must a vector whose complex expression is -5 - 5i be rotated until it coincides in direction with the vector whose complex expression is 3 + 4i?

—-> Please help me on this problem, I get confused on how the angles can be rotated.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The angles aren't being rotated; the vectors are.

For the complex number a+bi, the reference angle is ; then to get the actual angle you need to consider which quadrant the vector is in.

For the complex number -5-5i, the reference angle is arctan(-5/-5) = arctan(1), which is 45 degrees. With both components negative, the vector is in quadrant III, so the angle is 180+45 = 225 degrees.

The complex number 3+4i is in quadrant I, so the angle for 3+4i is

Find that angle; I'll call it x (degrees). To rotate the vector at 225 degrees to coincide with the vector at x degrees, you will "pass 0 degrees". So the calculation to find the angle through which the first vector is rotated to coincide with the second is (x+360)-225.

I leave the actual calculations to you.

Note that the first vector has a reference angle of 45 degrees in quadrant III and the second has a reference angle greater than 45 degrees in quadrant I, so the answer you come up with should be something slightly more than 180 degrees.

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