document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=161324>Question 161324</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is a value of a where vektors are (2,0,a), (1,a,1) and (2,1,a) make basis for R^3? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #118841 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=118841\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's call the vectors u,v, and w.<BR>\n" );
document.write( "u=(2,0,a)<BR>\n" );
document.write( "v=(1,a,1)<BR>\n" );
document.write( "w=(2,1,a)<BR>\n" );
document.write( "For the set to be a basis, their dot products must be zero (orthogonality condition).<BR>\n" );
document.write( "u*v=0<BR>\n" );
document.write( "v*w=0<BR>\n" );
document.write( "w*u=0<BR>\n" );
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document.write( "1.u*v=2*1+0*a+a*1=0<BR>\n" );
document.write( "1.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ba=0 ALIGN=MIDDLE  ALT=\"2%2Ba=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "2.v*w=1*2+a*1+1+a=0<BR>\n" );
document.write( "2.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2B2a=0 ALIGN=MIDDLE  ALT=\"2%2B2a=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "3.w*u=2*2+1*0+a*a=0<BR>\n" );
document.write( "3.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ba%5E2=0 ALIGN=MIDDLE  ALT=\"4%2Ba%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "From 1,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ba=0 ALIGN=MIDDLE  ALT=\"2%2Ba=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-2 ALIGN=MIDDLE  ALT=\"a=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This value for a also needs to satisfy eq. 2 and eq. 3,<BR>\n" );
document.write( "Checking with 2,<BR>\n" );
document.write( "2+2a=0<BR>\n" );
document.write( "2+2(-2)=-2<BR>\n" );
document.write( "the value a does not solve 2 and you can check that it doesn't solve 3 either.<BR>\n" );
document.write( "If the vectors formed a basis in R3, they must be orthogonal.<BR>\n" );
document.write( "From 1,2, and 3, you cannot find an \"a\" that solves all three equations.<BR>\n" );
document.write( "There is no solution.<BR>\n" );
document.write( "Additionally, you could have also concluded this since eq. 3 has no real solution.  </SPAN>\n" );
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