document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202170>Question 1202170</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the vector equation of the line that passes through the point A (-2, 3, 6) and is parallel to the line of intersection between the two planes\r<BR>\n" );
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document.write( "pi 1: 2x - y + z = 0 and<BR>\n" );
document.write( "pi 2: y  + 4z = 0\r<BR>\n" );
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document.write( "the answer is supposed to be vector r2 = (-2, 3, 6) + s (-5, -8, 2) sER </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836897 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=836897\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The given planes are<BR>\n" );
document.write( "2x - y + z = 0<BR>\n" );
document.write( "y + 4z = 0<BR>\n" );
document.write( "They intersect along some line which I'll call L1.\r<BR>\n" );
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document.write( "To determine the equation of a line, we need 2 points on it.\r<BR>\n" );
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document.write( "To generate a point in 2D, we plug in some x value to find y. This gives an (x,y) ordered pair.<BR>\n" );
document.write( "We'll do a similar thing in 3D for an ordered triple (x,y,z)\r<BR>\n" );
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document.write( "Let's plug in x = 0<BR>\n" );
document.write( "2x - y + z = 0<BR>\n" );
document.write( "2*0 - y + z = 0<BR>\n" );
document.write( "-y + z = 0\r<BR>\n" );
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document.write( "We have this system<BR>\n" );
document.write( "-y + z = 0<BR>\n" );
document.write( "y + 4z = 0\r<BR>\n" );
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document.write( "Apply the elimination method, or any other method of your choice, to find the solution to that system is (y,z) = (0,0)<BR>\n" );
document.write( "Coupled with x = 0 leads to the point (x,y,z) = (0,0,0) on the line L1.\r<BR>\n" );
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document.write( "-------------------------------------------------------\r<BR>\n" );
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document.write( "Let's try x = 1<BR>\n" );
document.write( "2x - y + z = 0<BR>\n" );
document.write( "2*1 - y + z = 0<BR>\n" );
document.write( "2 - y + z = 0<BR>\n" );
document.write( "-y + z = -2\r<BR>\n" );
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document.write( "We have this system<BR>\n" );
document.write( "-y + z = -2<BR>\n" );
document.write( "y + 4z = 0\r<BR>\n" );
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document.write( "Solve that system to get (y,z) = (8/5,-2/5)<BR>\n" );
document.write( "Therefore the point (1,8/5,-2/5) is also on the line L1.\r<BR>\n" );
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document.write( "The two points <BR>\n" );
document.write( "(0,0,0)<BR>\n" );
document.write( "and <BR>\n" );
document.write( "(1,8/5,-2/5)<BR>\n" );
document.write( "are on the line L1\r<BR>\n" );
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document.write( "The vector going from (0,0,0) to (1,8/5,-2/5) is < 1,8/5,-2/5 ><BR>\n" );
document.write( "This is a direction vector.<BR>\n" );
document.write( "Any parallel line will have the same direction vector or a scaled version of it.\r<BR>\n" );
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document.write( "One possible equation of line L1 as a vector equation is:<BR>\n" );
document.write( "(x,y,z) = startPoint + s*DirectionVector<BR>\n" );
document.write( "(x,y,z) = (-2,3,6) + s*(1,8/5,-2/5)\r<BR>\n" );
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document.write( "It may not be entirely clear how to go from the direction vector (1,8/5,-2/5) to (-5,-8,2), but we can introduce a scale factor.\r<BR>\n" );
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document.write( "Recall that vector (x,y,z) can be scaled to k*(x,y,z) = (kx,ky,kz) for any nonzero real number k.<BR>\n" );
document.write( "Vector (x,y,z) is parallel to vector (kx,ky,kz)\r<BR>\n" );
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document.write( "If we use k = -5, then<BR>\n" );
document.write( "k*(x,y,z) = (kx,ky,kz)<BR>\n" );
document.write( "k*(1,8/5,-2/5) = (k*1,k*8/5,k*(-2/5))<BR>\n" );
document.write( "-5*(1,8/5,-2/5) = (-5*1,-5*8/5,-5*(-2/5))<BR>\n" );
document.write( "-5*(1,8/5,-2/5) = (-5,-8,2)\r<BR>\n" );
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document.write( "The vectors (1,8/5,-2/5) and (-5,-8,2) are parallel. <BR>\n" );
document.write( "They point along the same straight line. \r<BR>\n" );
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document.write( "That is how we go from<BR>\n" );
document.write( "(x,y,z) = (-2,3,6) + s*(1,8/5,-2/5)<BR>\n" );
document.write( "to<BR>\n" );
document.write( "(x,y,z) = (-2,3,6) + s*(-5,-8,2)<BR>\n" );
document.write( "where \"s\" is any real number.<BR>\n" );
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