document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182669>Question 1182669</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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?\r<BR>\n" );
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document.write( "—-> Please help me on this problem, I get confused on how the angles can be rotated. </SPAN>\n" );
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document.write( "The angles aren't being rotated; the vectors are.<br><BR>\n" );
document.write( "For the complex number a+bi, the reference angle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%5E%28-1%29%28b%2Fa%29 ALIGN=MIDDLE  ALT=\"tan%5E%28-1%29%28b%2Fa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >; then to get the actual angle you need to consider which quadrant the vector is in.<br><BR>\n" );
document.write( "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.<br><BR>\n" );
document.write( "The complex number 3+4i is in quadrant I, so the angle for 3+4i is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%5E%28-1%29%284%2F3%29 ALIGN=MIDDLE  ALT=\"tan%5E%28-1%29%284%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "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.<br><BR>\n" );
document.write( "I leave the actual calculations to you.<br><BR>\n" );
document.write( "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.<br><BR>\n" );
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