document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=98494>Question 98494</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The question is<BR>\n" );
document.write( "Show that there are no vectors u and v such that the magnitude of u=1, the magbnitude of v=2, and the (dot product) u*v=3.<BR>\n" );
document.write( "Hope this makes sense. It is easier to understand with the symbols that they use. \r<BR>\n" );
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document.write( "I wasn't even sure how to start this problem. All I could get was 1*2 which doesn't =3 but that seems too easy.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #71659 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=71659\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'm not sure how far you are in your course, but to find the angle between any two vectors, you would use this 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{u}\cdot\v{v}}{\left|\v{u}\right|\left|\v{v}\right|}\"> where   <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left|\v{u}\right|\"> is the magnitude (ie length) of vector <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \v{u}\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left|\v{v}\right|\"> is the magnitude (ie length) of vector <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \v{v}\">\r<BR>\n" );
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document.write( "So let's assume that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \v{u}\cdot\v{v}=3\"> is true. That means we can replace <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \v{u}\cdot\v{v}\"> with 3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos(\theta)=\frac{3}{\left|\v{a}\right|\left|\v{b}\right|}\"> \r<BR>\n" );
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document.write( "Now plug in the given magnitudes <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left|\v{u}\right|=1\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left|\v{v}\right|=2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos(\theta)=\frac{3}{1*2}\"> \r<BR>\n" );
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document.write( "Now multiply the values in the denominator \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos(\theta)=\frac{3}{2}\"> \r<BR>\n" );
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document.write( "So this equation is implying that there is some value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta\"> that will make <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos(\theta)=\frac{3}{2}\"> true. However, there is no such value. Remember, the range of cosine is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -1\le cos(\theta)\le1\">. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2 ALIGN=MIDDLE  ALT=\"3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > equals 1.5 (which is greater than 1), it is outside the range of cosine. So if you try to take arccosine of both sides, you will not get a real answer for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta\">\r<BR>\n" );
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document.write( "So that means there are no two vectors that have magnitudes of 1 and 2 while having a dot product of 3.\r<BR>\n" );
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