document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=184328>Question 184328</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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:<BR>\n" );
document.write( "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?<BR>\n" );
document.write( "b) Suppose that a+bi is multiplied by &#8730;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?\r<BR>\n" );
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document.write( "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.\r<BR>\n" );
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document.write( "Please help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #138378 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=138378\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'll do the first part to get you started.\r<BR>\n" );
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document.write( "First, multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bbi ALIGN=MIDDLE  ALT=\"a%2Bbi\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Bi ALIGN=MIDDLE  ALT=\"1%2Bi\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bbi%29%281%2Bi%29=a%2Bai%2Bbi%2Bbi%5E2=a%2Bai%2Bbi-b ALIGN=MIDDLE  ALT=\"%28a%2Bbi%29%281%2Bi%29=a%2Bai%2Bbi%2Bbi%5E2=a%2Bai%2Bbi-b\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is what you have)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bbi%29%281%2Bi%29=%28a-b%29%2B%28a%2Bb%29i ALIGN=MIDDLE  ALT=\"%28a%2Bbi%29%281%2Bi%29=%28a-b%29%2B%28a%2Bb%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange and group like terms\r<BR>\n" );
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document.write( "So we essentially have two complex numbers <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bbi ALIGN=MIDDLE  ALT=\"a%2Bbi\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-b%29%2B%28a%2Bb%29i ALIGN=MIDDLE  ALT=\"%28a-b%29%2B%28a%2Bb%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If we plot these numbers on the complex plane, then we'll have the points (a,b) and (a-b,a+b)\r<BR>\n" );
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document.write( "Now think of these points as vectors\r<BR>\n" );
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document.write( "So we have two vectors <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \displaystyle \v{v_{1}}=<a,b>\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \displaystyle \v{v_{2}}=<a-b,a+b>\">\r<BR>\n" );
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document.write( "So the goal now is to find the angle between the two vectors (so we'll know the exact angle needed to rotate)\r<BR>\n" );
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document.write( "Remember, the angle between two vectors can be found through the formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos(\theta)=\frac{\v{a}\cdot\v{b}}{\left|\v{a}\right|\left|\v{b}\right|}\"> where   <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left|\v{a}\right|\"> is the magnitude (ie length) of vector <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \v{a}\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left|\v{b}\right|\"> is the magnitude (ie length) of vector <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \v{b}\">\r<BR>\n" );
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document.write( "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.\r<BR>\n" );
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document.write( "So let's calculate the dot product:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \v{v}_{1}\cdot\v{v}_{2}=(a)(a-b)+(b)(a+b)=a^2-ab+ab+b^2=a^2+b^2\">\r<BR>\n" );
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document.write( "Now let's calculate the magnitudes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left|\v{v}_{1}\right|=\left|<a,b>\right|=\sqrt{a^2+b^2}\"> \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left|\v{v}_{2}\right|=\left|<a-b,a+b>\right|=\sqrt{(a-b)^2+(a+b)^2}=\sqrt{a^2-2ab+b^2+a^2+2ab+b^2}=\sqrt{2a^2+2b^2}=\sqrt{2(a^2+b^2)}\">\r<BR>\n" );
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document.write( "Now multiply the magnitudes\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left|\v{v}_{1}\right|\cdot\left|\v{v}_{2}\right|=\sqrt{a^2+b^2}\cdot\sqrt{2(a^2+b^2)}=\sqrt{2(a^2+b^2)(a^2+b^2)}=\sqrt{2(a^2+b^2)^2}=\sqrt{2}\cdot\sqrt{(a^2+b^2)^2}=\sqrt{2}(a^2+b^2)\">\r<BR>\n" );
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document.write( "Now plug these values into the given formula above\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos(\theta)=\frac{a^2+b^2}{\sqrt{2}(a^2+b^2)}\">\r<BR>\n" );
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document.write( "Cancel out the common terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos(\theta)=\frac{1}{\sqrt{2}}\">\r<BR>\n" );
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document.write( "Now take the arccosine of both sides (to solve for theta)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=\arccos{\left(\frac{1}{\sqrt{2}}\right)}\">\r<BR>\n" );
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document.write( "Evaluate the arccosine of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fsqrt%282%29 ALIGN=MIDDLE  ALT=\"1%2Fsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F4 ALIGN=MIDDLE  ALT=\"pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > radians (or 45 degrees). Note: use the unit circle\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=\frac{\pi}{4}\"> or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=45\">\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the vector <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE <a,b>\"> needs to rotate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F4 ALIGN=MIDDLE  ALT=\"pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > radians (or 45 degrees) in order to be parallel to the vector <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE <a-b,a+b>\">\r<BR>\n" );
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document.write( "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 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2 ALIGN=MIDDLE  ALT=\"a=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=3 ALIGN=MIDDLE  ALT=\"b=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So the first number would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2B3i ALIGN=MIDDLE  ALT=\"2%2B3i\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the second number would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282-3%29%2B%282%2B3%29i=-1%2B5i ALIGN=MIDDLE  ALT=\"%282-3%29%2B%282%2B3%29i=-1%2B5i\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now find the angle between the vectors <2,3> and <-1,5> and you'll find that it will be 45 degrees. </SPAN>\n" );
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