document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=364482>Question 364482</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi,<BR>\n" );
document.write( "Im Struggeling with this question:\r<BR>\n" );
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document.write( "Vectors:\r<BR>\n" );
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document.write( "Show that vectors a = 2i-4j-k, b = 3i+2j-2k and c = 5i-2j-3k can form the sides of a triangle, find the length of each side and hence show that the triangle has a right angle. sketch the triangle of vectors, indicating their directions and the right angle,\r<BR>\n" );
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document.write( "Any help would be much appreciated?<BR>\n" );
document.write( "Thanks<BR>\n" );
document.write( "Scott </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #759805 by </B><A HREF=/tutors/aboutme.mpl?userid=isku><B>isku(1)</B></A><img src=\"/images/stars/iconNewId_16x16.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=isku><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=759805\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If two of three unit vectors are the same then they cannot  form a triangle.\r<BR>\n" );
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document.write( "|a| = sqrt(4+16+1) = sqrt(21)       <-length of a <BR>\n" );
document.write( "|b| = sqrt(9+4+4 ) = sqrt(17)       <-length of b<BR>\n" );
document.write( "|c| = sqrt(25+4+9) = sqrt(38)       <-length of c\r<BR>\n" );
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document.write( "â = a/|a|     <-unit vector for a<BR>\n" );
document.write( "   = [ 2, -4, -1  ]/ sqrt( 21) <BR>\n" );
document.write( "b̂  = b/|b|     <-unit vector for b<BR>\n" );
document.write( "     = [ 3, 2, -2  ]/sqrt(17)<BR>\n" );
document.write( "ĉ  = c/|c|     <-unit vector for c<BR>\n" );
document.write( "    = [ 5, -2, -3  ]/ sqrt(38)\r<BR>\n" );
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document.write( "If two vector are the same then the  difference is zero<BR>\n" );
document.write( "â -  b̂  = [ 2, -4, -1  ]/ sqrt( 21) - [ 3, 2, -2  ]/sqrt(17) ≠ [0 0 0]  <--they are not the same <BR>\n" );
document.write( "â -  ĉ = [ 2, -4, -1  ]/ sqrt( 21) - [ 5, -2, -3  ]/ sqrt(38) ≠ [0 0 0] <--they are not the same <BR>\n" );
document.write( "b̂ - ĉ  = [ 3, 2, -2  ]/sqrt(17) - [ 5, -2, -3  ]/ sqrt(38) ≠ [0 0 0] <--they are not the same\r<BR>\n" );
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document.write( "To control the assumption, let d = [ 2*2, -4*2, -1*2  ]    i.e. 2*a<BR>\n" );
document.write( "d̂ = [ 2*2, -4*2, -1*2  ]  / sqrt( 16+64+4) = [ 2*2, -4*2, -1*2  ]  / sqrt(84)<BR>\n" );
document.write( "â -  d̂  = [ 2, -4, -1  ]/ sqrt( 21) - [ 2*2, -4*2, -1*2  ]  / sqrt(84) = [0 0 0] → â and  d̂  are the same. <BR>\n" );
document.write( "→ We cannot form a triangle by these three :  a,b and d or  a,c and d\r<BR>\n" );
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document.write( "Showing that it is right-angled<BR>\n" );
document.write( "|a| = sqrt(4+16+1) = sqrt(21)    length of a <BR>\n" );
document.write( "|b| = sqrt(9+4+4 ) = sqrt(17)    length of b<BR>\n" );
document.write( "|c| = sqrt(25+4+9) = sqrt(38)    length of c\r<BR>\n" );
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document.write( "Since |c|^2 = |a|^2 + |b|^2      ( using Pythagoras) <BR>\n" );
document.write( "→ they forms a right-angled. \r<BR>\n" );
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document.write( " </SPAN>\n" );
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