document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203665>Question 1203665</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that 4u · v = ||u + v||^2 − ||u − v||^2. Show that u and v is orthogonal if and<BR>\n" );
document.write( "only if ||u + v|| = ||u − v||.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #839413 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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/cgi-bin/show-face.mpl?user=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=839413\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Show that \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4u+%2A+v+=+abs%28abs%28u+%2B+v%29%29%5E2+%E2%88%92+abs%28abs%28u+%E2%88%92+v%29%29%5E2 ALIGN=MIDDLE  ALT=\"4u+%2A+v+=+abs%28abs%28u+%2B+v%29%29%5E2+%E2%88%92+abs%28abs%28u+%E2%88%92+v%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Hint. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28abs%28u%29%29%5E2=u%2Au ALIGN=MIDDLE  ALT=\"abs%28abs%28u%29%29%5E2=u%2Au\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28u%2Bv%29%5E2-%28u-v%29%5E2 ALIGN=MIDDLE  ALT=\"%28u%2Bv%29%5E2-%28u-v%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28u%2Bv%29%28u%2Bv%29-%28u-v%29%28u-v%29 ALIGN=MIDDLE  ALT=\"%28u%2Bv%29%28u%2Bv%29-%28u-v%29%28u-v%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%5E2+%2B+2uv+%2B+v%5E2-%28u%5E2+-+2uv+%2B+v%5E2%29 ALIGN=MIDDLE  ALT=\"u%5E2+%2B+2uv+%2B+v%5E2-%28u%5E2+-+2uv+%2B+v%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%5E2+%2B+2uv+%2B+v%5E2-u%5E2+%2B2uv+-+v%5E2 ALIGN=MIDDLE  ALT=\"u%5E2+%2B+2uv+%2B+v%5E2-u%5E2+%2B2uv+-+v%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%28u%5E2%29+%2B+2uv+%2B+cross%28v%5E2%29-cross%28u%5E2%29+%2B2uv+-+cross%28v%5E2%29 ALIGN=MIDDLE  ALT=\"cross%28u%5E2%29+%2B+2uv+%2B+cross%28v%5E2%29-cross%28u%5E2%29+%2B2uv+-+cross%28v%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2uv++%2B2uv+ ALIGN=MIDDLE  ALT=\"2uv++%2B2uv+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4uv+ ALIGN=MIDDLE  ALT=\"4uv+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Show that u and v is orthogonal if and only if <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28abs%28u+%2B+v%29%29+=+abs%28abs%28u-v%29%29 ALIGN=MIDDLE  ALT=\"abs%28abs%28u+%2B+v%29%29+=+abs%28abs%28u-v%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u ALIGN=MIDDLE  ALT=\"u\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v ALIGN=MIDDLE  ALT=\"v\"   DISABLED_style= \"max-width:90%; height:auto;\" > are orthogonal vectors, then:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28abs%28u%2Bv%29%29%5E2=abs%28abs%28u%29%29%5E2%2Babs%28abs%28v%29%29%5E2 ALIGN=MIDDLE  ALT=\"abs%28abs%28u%2Bv%29%29%5E2=abs%28abs%28u%29%29%5E2%2Babs%28abs%28v%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28abs%28u-v%29%29%5E2=abs%28abs%28u%29%29%5E2%2Babs%28abs%28-v%29%29%5E2=abs%28abs%28u%29%29%5E2%2Babs%28abs%28v%29%29%5E2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28abs%28u%2Bv%29%29%5E2=abs%28abs%28u-v%29%29%5E2 ALIGN=MIDDLE  ALT=\"abs%28abs%28u%2Bv%29%29%5E2=abs%28abs%28u-v%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but the norm is ever positive therefore: \r<BR>\n" );
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document.write( "now, if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28abs%28u%2Bv%29%29=abs%28abs%28u-v%29%29 ALIGN=MIDDLE  ALT=\"abs%28abs%28u%2Bv%29%29=abs%28abs%28u-v%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we have:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28abs%28u%2Bv%29%29%5E2=abs%28abs%28u%29%29%5E2%2B2u%2Av%2Babs%28abs%28v%29%29%5E2 ALIGN=MIDDLE  ALT=\"abs%28abs%28u%2Bv%29%29%5E2=abs%28abs%28u%29%29%5E2%2B2u%2Av%2Babs%28abs%28v%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28abs%28u-v%29%29%5E2=abs%28abs%28u%29%29%5E2-2u%2Av%2Babs%28abs%28v%29%29%5E2 ALIGN=MIDDLE  ALT=\"abs%28abs%28u-v%29%29%5E2=abs%28abs%28u%29%29%5E2-2u%2Av%2Babs%28abs%28v%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "By the equality\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28abs%28u%29%29%5E2%2B2u%2Av%2Babs%28abs%28v%29%29%5E2=abs%28abs%28u%29%29%5E2-2u%2Av%2Babs%28abs%28v%29%29%5E2+if+and+only+if%0D%0A%0D%0A%7B%7B%7B2u%2Av=-2u%2Av ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <=> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4u%2Av=0 ALIGN=MIDDLE  ALT=\"4u%2Av=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%2Av=0 ALIGN=MIDDLE  ALT=\"u%2Av=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "this means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u ALIGN=MIDDLE  ALT=\"u\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v ALIGN=MIDDLE  ALT=\"v\"   DISABLED_style= \"max-width:90%; height:auto;\" > are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=orthogonal ALIGN=MIDDLE  ALT=\"orthogonal\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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