document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1042853>Question 1042853</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Calculate proj v to u. u=<2,9> and v= (-3,4)\r<BR>\n" );
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document.write( "Resolve u into u1 and u2, where u1 is parallel to v and u2 is orthogonal to v. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #658062 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=658062\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The projection of v onto u is given by \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28u%2Av%29%2Fabs%28u%29%5E2%29u+=+%28%28-6%2B36%29%2F%284%2B81%29%29u+=+%2830%2F85%29u+=+%286%2F17%29u ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "First find vector <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%5B1%5D ALIGN=MIDDLE  ALT=\"u%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is the component of vector u along vector v.\r<BR>\n" );
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document.write( "This is given by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28u%2Av%29%2Fabs%28v%29%5E2%29v+=+%28%28-6%2B36%29%2F%289%2B16%29%29v+=+%286%2F5%29v ALIGN=MIDDLE  ALT=\"%28%28u%2Av%29%2Fabs%28v%29%5E2%29v+=+%28%28-6%2B36%29%2F%289%2B16%29%29v+=+%286%2F5%29v\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  This is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%5B1%5D ALIGN=MIDDLE  ALT=\"u%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now let's find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%5B2%5D ALIGN=MIDDLE  ALT=\"u%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  A vector orthogonal (perpendicular) to v = <-3,4> is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5Bo%5D+ ALIGN=MIDDLE  ALT=\"v%5Bo%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <4,3>, since their dot product is -3*4 + 4*3 = 0.  \r<BR>\n" );
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document.write( "Next find the component of u along <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5Bo%5D+ ALIGN=MIDDLE  ALT=\"v%5Bo%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  <BR>\n" );
document.write( "It is given by \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28u%2Av%5Bo%5D%29%2Fabs%28v%5Bo%5D%29%5E2%29v%5Bo%5D+=+%28%288%2B27%29%2F%2816%2B9%29%29v%5Bo%5D+=+%287%2F5%29v%5Bo%5D ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "This is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%5B2%5D ALIGN=MIDDLE  ALT=\"u%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Since it is easy to verify that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%5B1%5D%2Bu%5B2%5D+=+%286%2F5%29v+%2B+%287%2F5%29v%5Bo%5D+=+u ALIGN=MIDDLE  ALT=\"u%5B1%5D%2Bu%5B2%5D+=+%286%2F5%29v+%2B+%287%2F5%29v%5Bo%5D+=+u\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we have found the resolution of u into its orthogonal components. </SPAN>\n" );
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