document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1049379>Question 1049379</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1)In 3d-space, let v1=(0,2,1) v2=(2,1,1) v3=(1,3,-1) v4=(4,2,1). Write v4 as linear combination of v1,v2,v3.\r<BR>\n" );
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document.write( "2)let v1=(0,-4,7) and v2=(3,-2,-1)<BR>\n" );
document.write( "find<BR>\n" );
document.write( "i)length of v1<BR>\n" );
document.write( "ii) v1+v2<BR>\n" );
document.write( "iii)length of v1+v2<BR>\n" );
document.write( "iv)angle between v1 and v2<BR>\n" );
document.write( "v)v1.v2<BR>\n" );
document.write( "vi)v1xv2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #664932 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=664932\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1.(4,2,1)=a(0,2,1)+b(2,1,1)+c(1,3,-1)<BR>\n" );
document.write( "(4,2,1)=(0a+2b+c,2a+b+3c,a+b-c)<BR>\n" );
document.write( "Using Cramer's rule,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%283%2C3%2C%0D%0A0%2C2%2C1%2C%0D%0A2%2C1%2C3%2C%0D%0A1%2C1%2C-1%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C%0D%0A0%2C2%2C1%2C%0D%0A2%2C1%2C3%2C%0D%0A1%2C1%2C-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%5Ba%5D%29=-5 ALIGN=MIDDLE  ALT=\"abs%28A%5Ba%5D%29=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%5Bb%5D%29=20 ALIGN=MIDDLE  ALT=\"abs%28A%5Bb%5D%29=20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%5Bc%5D%29=4 ALIGN=MIDDLE  ALT=\"abs%28A%5Bc%5D%29=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=abs%28A%5Ba%5D%29%2Fabs%28A%29=-5%2F11 ALIGN=MIDDLE  ALT=\"a=abs%28A%5Ba%5D%29%2Fabs%28A%29=-5%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=abs%28A%5Bb%5D%29%2Fabs%28A%29=20%2F11 ALIGN=MIDDLE  ALT=\"b=abs%28A%5Bb%5D%29%2Fabs%28A%29=20%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=abs%28A%5Bc%5D%29%2Fabs%28A%29=4%2F11 ALIGN=MIDDLE  ALT=\"c=abs%28A%5Bc%5D%29%2Fabs%28A%29=4%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So then,<BR>\n" );
document.write( "(4,2,1)=(-5/11)*(0,2,1)+(20/11)*(2,1,1)+(4/11)*(1,3,-1)<BR>\n" );
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document.write( "2.i) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28V%5B1%5D%29=sqrt%280%5E2%2B%28-4%29%5E2%2B7%5E2%29 ALIGN=MIDDLE  ALT=\"abs%28V%5B1%5D%29=sqrt%280%5E2%2B%28-4%29%5E2%2B7%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "ii) V[1]+V[2]=(0+3,-4-2,7-1)=(3,-6,6)<BR>\n" );
document.write( "iii)<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28V%5B1%5D%2BV%5B2%5D%29=sqrt%283%5E2%2B%28-6%29%5E2%2B6%5E2%29 ALIGN=MIDDLE  ALT=\"abs%28V%5B1%5D%2BV%5B2%5D%29=sqrt%283%5E2%2B%28-6%29%5E2%2B6%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "iv) Use the dot product,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5B1%5D%2AV%5B2%5D=0%2A3%2B%28-4%29%28-2%29%2B7%28-1%29=8-7=1 ALIGN=MIDDLE  ALT=\"V%5B1%5D%2AV%5B2%5D=0%2A3%2B%28-4%29%28-2%29%2B7%28-1%29=8-7=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29=%28V%5B1%5D%2AV%5B2%5D%29%2F%28abs%28V%5B1%5D%29abs%28V%5B2%5D%29%29 ALIGN=MIDDLE  ALT=\"cos%28theta%29=%28V%5B1%5D%2AV%5B2%5D%29%2F%28abs%28V%5B1%5D%29abs%28V%5B2%5D%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28V%5B2%5D%29=sqrt%283%5E2%2B%28-2%29%5E2%2B%28-1%29%5E2%29 ALIGN=MIDDLE  ALT=\"abs%28V%5B2%5D%29=sqrt%283%5E2%2B%28-2%29%5E2%2B%28-1%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Plug in the values to get the answer.<BR>\n" );
document.write( "v) From previous, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5B1%5D%2AV%5B2%5D=1 ALIGN=MIDDLE  ALT=\"V%5B1%5D%2AV%5B2%5D=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "vi)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5B1%5DxV%5B2%5D=%28matrix%283%2C3%2C%0D%0Ai%2Cj%2Ck%2C%0D%0A0%2C-4%2C7%2C%0D%0A3%2C-2%2C-1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5B1%5DxV%5B2%5D=%28matrix%281%2C3%2C%0D%0A18%2C21%2C12%29%29 ALIGN=MIDDLE  ALT=\"V%5B1%5DxV%5B2%5D=%28matrix%281%2C3%2C%0D%0A18%2C21%2C12%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );