document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=148541>Question 148541</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help.<BR>\n" );
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document.write( "find the parametric equations for line passing through the points P(5,-2,4)     Q(7,2,-4).<BR>\n" );
document.write( "#2<BR>\n" );
document.write( "show that if v is a nonzero vector, then v/ llvll is a unit vector<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #108916 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=108916\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> #1<BR>\n" );
document.write( "find the parametric equations for line passing through the points P(5,-2,4) Q(7,2,-4).<BR>\n" );
document.write( "#2<BR>\n" );
document.write( "show that if v is a nonzero vector, then v/ llvll is a unit vector<BR>\n" );
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document.write( "#1<BR>\n" );
document.write( "Using determinants:\r<BR>\n" );
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document.write( "x  y  z  1<BR>\n" );
document.write( "5 -2  4  1<BR>\n" );
document.write( "7  2 -4  1\r<BR>\n" );
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document.write( "gives:<BR>\n" );
document.write( "2y + z = 0<BR>\n" );
document.write( "This can be expressed in parametric terms. Use t for the parameter (time, eg)<BR>\n" );
document.write( "y = t<BR>\n" );
document.write( "z = -2t<BR>\n" );
document.write( "Other coefficients are usable, as long as the ratio is maintained.<BR>\n" );
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document.write( "#2 show that if v is a nonzero vector, then v/ llvll is a unit vector<BR>\n" );
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document.write( "I'll edit later and add this.<BR>\n" );
document.write( "Vector v is general, connecting 2 points in space. It can be moved (translated) so that one end is at the origin (0,0,0) and the other end is the point (x,y,z).<BR>\n" );
document.write( "The length of the vector is sqrt(x^2+y^2+z^2).  This is also the absolute value of the vector.<BR>\n" );
document.write( "That's proof that dividing the vector by its abs value gives a unit vector, but in math terms:<BR>\n" );
document.write( "v/(ABS(v))= x/ABS + y/ABS + z/ABS<BR>\n" );
document.write( "= x/sqrt() + y/sqrt() + z/sqrt()<BR>\n" );
document.write( "Now calculate the length<BR>\n" );
document.write( "= sqrt{ [x/sqrt()]^2 + [y/sqrt()]^2 + [z/sqrt()]^2}<BR>\n" );
document.write( "= sqrt{ (x^2+y^2+z^2)/(x^2+y^2+z^2) } (DEN = ABS value from above)<BR>\n" );
document.write( "= 1. So it's a unit vector parallel to the original.<BR>\n" );
document.write( "This is similar to proving that 2 = 2, imo.\r<BR>\n" );
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