document.write( "Question 1201894: Find the general form of the equation of the plane passing through the points
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Algebra.Com's Answer #836458 by ikleyn(52794) $\"\"$ $\"About$

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\n" ); document.write( "Find the general form of the equation of the plane passing through the points
\n" ); document.write( "A (3,2,2), B (1,5,0), and C (1,-3,1).
\n" ); document.write( "Use point A as 21, Y, 21 in the equation of the plane.
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\n" ); document.write( "\n" ); document.write( " The instruction in the last line of your post is non-sensical, so I ignore it.\r
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document.write( "The solution is in two steps.\r\n" );
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document.write( "                   Step 1 - making equation of the plane parallel to the desired plane \r\n" );
document.write( "                            and passing through the origin of the coordinate system\r\n" );
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document.write( "Form vector AB_hat from point A to point B\r\n" );
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document.write( "     AB_hat = B-A = (1-3,5-2,0-2) = (-2,3,-2).\r\n" );
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document.write( "Form vector AC_hat from point A to point C\r\n" );
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document.write( "     AC_hat = C-A = (1-3,-3-2,1-2) = (-2,-5,-1).\r\n" );
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document.write( "Make 3x3-matrix\r\n" );
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document.write( "     M = .\r\n" );
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document.write( "In this matrix, x, y and z are variables representing coordinates along x-, y- and z-axis;\r\n" );
document.write( "the second row contains the components of vector AB_hat;\r\n" );
document.write( "the third  row contains the components of vector AC_hat.\r\n" );
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document.write( "Write the determinant of matrix M, expanding the determinant by standard rules\r\n" );
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document.write( "    det (M) = x*(3*(-1)-(-5)*(-2)) - y*((-2)*(-1)-(-2)*(-2)) + z*((-2)*(-5)-(-2)*3) = -13x + 2y + 16z.\r\n" );
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document.write( "This equation \r\n" );
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document.write( "       -13x + 2y + 16z = 0     (1)\r\n" );
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document.write( "determines the plane, parallel to the desired plane and passing  through the origin of the coordinate system.\r\n" );
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document.write( "                               Step 1 is complete.\r\n" );
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document.write( "                   Step 2 - making equation of the desired plane\r\n" );
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document.write( "The desired plane has an equation of the form\r\n" );
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document.write( "       -13x + 2y + 16z = d.    (2)\r\n" );
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document.write( "Its left side is the same as in equation (1).\r\n" );
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document.write( "Right side \"d\" is a real value which we need to determine.\r\n" );
document.write( "This value of \"d\" is the value of the linear function in the left side of equation (2),\r\n" );
document.write( "calculated at any of the points A, or B, or C.\r\n" );
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document.write( "The interesting fact (actually, a key fact) is that the value of this function is THE SAME \r\n" );
document.write( "for all points A, B and C:\r\n" );
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document.write( "    for point A, it is  (-13)*3 + 2*2    + 16*2 = -3;\r\n" );
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document.write( "    for point B, it is  (-13)*1 + 2*5    + 16*0 = -3;\r\n" );
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document.write( "    for point C, it is  (-13)*1 + 2*(-3) + 16*1 = -3.\r\n" );
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document.write( "Thus finally, the desired equation is \r\n" );
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document.write( "    -13x + 2y + 16z = -3.     ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "In this post, I showed you the technique of calculations,
\n" ); document.write( "based on the standard algorithm for the given points.\r
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\n" ); document.write( "\n" ); document.write( "Why the algorithm works this way, is described in any course of analytic geometry
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\n" ); document.write( "\n" ); document.write( "They are the necessary pre-requisites for such problems, so, it is assumed
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\n" ); document.write( "\n" ); document.write( "In my post, I only exposed how this algorithm works in shortest form,
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\n" ); document.write( "\n" ); document.write( "Using this post, you can solve any other similar problem, simply following this TEMPLATE.\r
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