document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=20758>Question 20758</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Recall that P3 is the space of all polynomials of degree less than three with real coefficients.  Define an inner product < , > on P3 given by:<BR>\n" );
document.write( "  < p,q > = the sum from i=1 to 3 of p(i-1)*q(i-1), and let ||  || be the norm with respect to this inner product.  <BR>\n" );
document.write( "(a) let p(x)=1+x-x^2 and q(x)=2-x<BR>\n" );
document.write( "Find a polynomial f(x) in P3 which is orthogonal to both p(x) and q(x) with respect to the inner product.<BR>\n" );
document.write( "(b) find the distance between p(x)+q(x) and f(x).<BR>\n" );
document.write( "thank you so much for your help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #10091 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><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=10091\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Recall that P3 is the space of all polynomials of degree less than three with real coefficients. Define an inner product < , > on P3 given by:<BR>\n" );
document.write( "< p,q > = the sum from i=1 to 3 of p(i-1)*q(i-1), and let || || be the norm with respect to this inner product. <BR>\n" );
document.write( "(a) let p(x)=1+x-x^2 and q(x)=2-x<BR>\n" );
document.write( "Find a polynomial f(x) in P3 which is orthogonal to both p(x) and q(x) with respect to the inner product.<BR>\n" );
document.write( "(b) find the distance between p(x)+q(x) and f(x).\r<BR>\n" );
document.write( "\n" );
document.write( " [Note <p,q> as regular dot product in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E3 ALIGN=MIDDLE  ALT=\"R%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >]\r<BR>\n" );
document.write( "\n" );
document.write( " Sol: a) Let f = pxq (as cross product)<BR>\n" );
document.write( "             = |i j k|  (i as 1, j as x, k as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "               |1 1 -1|<BR>\n" );
document.write( "               |2-1 0|<BR>\n" );
document.write( "            = -i - 2j + -3k = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+-+2x+-+3x%5E2 ALIGN=MIDDLE  ALT=\"-1+-+2x+-+3x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "        Or<BR>\n" );
document.write( "        Let f = a  + bx + cx^2 in P3, if <f,p> = <f,q> = 0, then<BR>\n" );
document.write( "         <f,p> = (a,b,c). (1,1,-1) = a+b-c = 0 &<BR>\n" );
document.write( "         <f,q> = (a,b,c). (2,-1,0) = 2a-b = 0<BR>\n" );
document.write( "         then we have b=2a, c= a+b= 3a.<BR>\n" );
document.write( "         f = a (1+2x + 3x^2) for some a.<BR>\n" );
document.write( "     b) Let f(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+-+2x+-+3x%5E2 ALIGN=MIDDLE  ALT=\"-1+-+2x+-+3x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "||p(x)+q(x)- f(x)|| = || (3-x^2) + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+2x+%2B+3x%5E2 ALIGN=MIDDLE  ALT=\"1+%2B+2x+%2B+3x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >||<BR>\n" );
document.write( " = || <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+%2B2x+%2B2x%5E2 ALIGN=MIDDLE  ALT=\"4+%2B2x+%2B2x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >||<BR>\n" );
document.write( " = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sqrt%284%5E2%2B+2%5E2+%2B+2%5E2%29 ALIGN=MIDDLE  ALT=\"+sqrt%284%5E2%2B+2%5E2+%2B+2%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A+sqrt%286%29 ALIGN=MIDDLE  ALT=\"2%2A+sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " Kenny<BR>\n" );
document.write( " PS: 1.Sorry for answering you late.<BR>\n" );
document.write( "     2.It seems that you posted this question in wrong forum.<BR>\n" );
document.write( "     3.You have to work hard. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );