document.write( "Question 1203102: Determine whether the lines L1 and L2 are parallel, skew, or intersecting.\r
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\n" ); document.write( "\n" ); document.write( "L1: x = 6+4t, y = 8-2t, z = 2+6t
\n" ); document.write( "L2: x = 4+16s, y = 12-8s, z = 16+20s
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Algebra.Com's Answer #838391 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Determine whether the lines L1 and L2 are parallel, skew, or intersecting.\r
\n" ); document.write( "\n" ); document.write( "L1: x = 6+4t, y = 8-2t, z = 2+6t
\n" ); document.write( "L2: x = 4+16s, y = 12-8s, z = 16+20s
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document.write( "Consider the guiding vectors of these lines.\r\n" );
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document.write( "The guiding vectors are comprised of the coefficients of parametric equations.\r\n" );
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document.write( "For line L1 the guiding vector is  ( 4,-2,6);\r\n" );
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document.write( "For line L2 the guiding vector is  (16,-8,20).\r\n" );
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document.write( "It is clearly seen that the guiding vectors are not proportional - hence, lines L1 and L2 are not parallel.\r\n" );
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document.write( "To find out, if the lines L1 and L2 do intersect, you should consider this system of 3 equations in 2 unknowns\r\n" );
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document.write( "    6 + 4t =  4 + 16s,     (1)\r\n" );
document.write( "    8 - 2t = 12 -  8s,     (2)\r\n" );
document.write( "    2 + 6t = 16 + 20t.     (3)\r\n" );
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document.write( "Multiply equation (2) by 2 (both sides).  Keep equation (1) as is.  You will get\r\n" );
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document.write( "    6 + 4t =  4 + 16s,     (1')\r\n" );
document.write( "   16 - 4t = 24 - 16s.     (2')\r\n" );
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document.write( "Add equations (1) and (2').  The terms with \"t\" will cancel each other; the terms with \"s\" will cancel each other, too.\r\n" );
document.write( "Thus, you will get a self-contradictory equality\r\n" );
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document.write( "      22   =    28.\r\n" );
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document.write( "It means that the system of equations (1), (2) has no solutions (is inconsistent).\r\n" );
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document.write( "From it, we conclude that the system of equations (1), (2), (3) has no solutions.\r\n" );
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document.write( "So, lines L1 and L2 are skew lines in 3D : they are not parallel and are not intersecting.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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