document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1044123>Question 1044123</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Is the following statement always, sometimes or never true? Justify your reasoning\r<BR>\n" );
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document.write( "there should be arrows above the letters\r<BR>\n" );
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document.write( " |u + v| > |u - v|\r<BR>\n" );
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document.write( "thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #659375 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=659375\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Let's set up some notation<BR>\n" );
document.write( "a,b,c,d are scalars<BR>\n" );
document.write( "u and v are vectors such that<BR>\n" );
document.write( "u = (a,b)<BR>\n" );
document.write( "v = (c,d)\r<BR>\n" );
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document.write( "So that means<BR>\n" );
document.write( "u+v = (a+c,b+d)<BR>\n" );
document.write( "u-v = (a-c,b-d)\r<BR>\n" );
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document.write( "Now use the length of vector formula to get<BR>\n" );
document.write( "|u + v| = sqrt( (a+c)^2 + (b+d)^2 )<BR>\n" );
document.write( "|u + v| = sqrt( a^2+2ac+c^2 + b^2+2bd+d^2 )\r<BR>\n" );
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document.write( "and this as well<BR>\n" );
document.write( "|u - v| = sqrt( (a-c)^2 + (b-d)^2 )<BR>\n" );
document.write( "|u - v| = sqrt( a^2-2ac+c^2 + b^2-2bd+d^2 )\r<BR>\n" );
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document.write( "Now let's go back to <BR>\n" );
document.write( "|u + v| > |u - v|\r<BR>\n" );
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document.write( "Perform substitutions and then square both sides to get<BR>\n" );
document.write( "|u + v| > |u - v|<BR>\n" );
document.write( "sqrt( a^2+2ac+c^2 + b^2+2bd+d^2 ) > sqrt( a^2-2ac+c^2 + b^2-2bd+d^2 )<BR>\n" );
document.write( "a^2+2ac+c^2 + b^2+2bd+d^2 > a^2-2ac+c^2 + b^2-2bd+d^2\r<BR>\n" );
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document.write( "Things look messy, but we can subtract a^2, c^2, b^2 and d^2 from both sides to have those terms cancel out. We will be left with this\r<BR>\n" );
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document.write( "2ac + 2bd > -2ac - 2bd\r<BR>\n" );
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document.write( "add 2ac and 2bd to both sides and you'll get\r<BR>\n" );
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document.write( "2ac+2bd+2ac+2bd > 0<BR>\n" );
document.write( "4ac+4bd > 0\r<BR>\n" );
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document.write( "which ultimately simplifies to ac+bd > 0\r<BR>\n" );
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document.write( "-------------------------------------------------------\r<BR>\n" );
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document.write( "So if |u+v| > |u-v|, where u and v are defined above, then ac+bd > 0. The same can be said in reverse as well. \r<BR>\n" );
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document.write( "There are some cases where ac+bd > 0 is true. But it could be false as well. \r<BR>\n" );
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document.write( "For instance, if a = 2, b = 3, c = 1 and d = 2, then the inequality would be true.\r<BR>\n" );
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document.write( "Compare this to if a = 2, b = 3, c = -1 and d = -2, then the inequality would be false. \r<BR>\n" );
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document.write( "Side note: the inequality is only true if the angle between the vectors u and v is acute. If the angle is obtuse, then the inequality is false.\r<BR>\n" );
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document.write( "So the final answer is <font color=red size=4>sometimes</font><BR>\n" );
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