document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201853>Question 1201853</A>: <I><SPAN STYLE=\"word-break: break-all;\"> this is from my 'Cartesian Equation of a Plane' lesson.\r<BR>\n" );
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document.write( "a. Determine the Cartesian equation of the plane that contains the three points A(-2,3,1), B(3,4,5) and C(1,1,0).\r<BR>\n" );
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document.write( "b. Determine the Cartesian equation of the plane containing the point P(-1,1,0) and perpendicular to the line between the points A(1,2,1) and B(3,-2,0). \r<BR>\n" );
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document.write( "thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836451 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</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=836451\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answers:<BR>\n" );
document.write( "(a) <font color=red size=4>7x+17y-13z  = 24</font><BR>\n" );
document.write( "(b) <font color=red size=4>2x-4y-z = -6</font><BR>\n" );
document.write( "Other answers are possible.\r<BR>\n" );
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document.write( "Explanation for part (a)\r<BR>\n" );
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document.write( "We have these three points<BR>\n" );
document.write( "A(-2,3,1)<BR>\n" );
document.write( "B(3,4,5)<BR>\n" );
document.write( "C(1,1,0)\r<BR>\n" );
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document.write( "Subtract the coordinates of B and A to find the vector that points from A to B.<BR>\n" );
document.write( "B-A = < 3,4,5 > - < -2,3,1 ><BR>\n" );
document.write( "B-A = < 3,4,5 > + < 2,-3,-1 ><BR>\n" );
document.write( "B-A = < 3+2,4-3,5-1 ><BR>\n" );
document.write( "B-A = < 5,1,4 ><BR>\n" );
document.write( "This says to go from A to B, we do three things:<ul><li>Move 5 units along the positive x axis.</li><li>Move 1 unit along the positive y axis.</li><li>Move 4 units along the positive z axis.</li></ul>Therefore, vector AB is < 5,1,4 ><BR>\n" );
document.write( "The order in vector naming is important. \"Vector AB\" means we start at A and point to B.<BR>\n" );
document.write( "While \"vector BA\" means we start at B and point at A.\r<BR>\n" );
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document.write( "Repeat similar steps to find vector AC = < 3,-2,-1 >\r<BR>\n" );
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document.write( "We found that<BR>\n" );
document.write( "vector AB = < 5,1,4 ><BR>\n" );
document.write( "vector AC = < 3,-2,-1 >\r<BR>\n" );
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document.write( "Next, take the cross product of these two vectors. <BR>\n" );
document.write( "This will construct a vector perpendicular to both AB and AC, <BR>\n" );
document.write( "Think of this new vector as a vertical pole out of the flat horizontal ground. Vectors AB and AC are entirely on the flat ground.<BR>\n" );
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document.write( "The cross product of vectors AB and AC is < 7, 17, -13 > as discussed in this lesson here<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.algebra.com/algebra/homework/Vectors/cross-product.lesson\">https://www.algebra.com/algebra/homework/Vectors/cross-product.lesson</a><BR>\n" );
document.write( "This represents the normal vector to the plane.<BR>\n" );
document.write( "We can think of the normal vector being of the form < a,b,c ><BR>\n" );
document.write( "In this case < a,b,c > = < 7,17,-13 ><BR>\n" );
document.write( "The normal vector tells us how to tilt the plane.\r<BR>\n" );
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document.write( "One template equation for a plane is<BR>\n" );
document.write( "a(x - p) + b(y - q) + c(z - r) = 0<BR>\n" );
document.write( "where<BR>\n" );
document.write( "a,b,c = coordinates of the normal vector we just found<BR>\n" );
document.write( "p,q,r = coordinates of a point that is on the plane\r<BR>\n" );
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document.write( "We have 3 choices for what we pick for p,q,r<BR>\n" );
document.write( "I'll go for (p,q,r) = (-2,3,1) which is the location of point A.<BR>\n" );
document.write( "You could pick the coordinates of B or C.\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "a(x-p) + b(y-q) + c(z - r) = 0<BR>\n" );
document.write( "7(x-p) + 17(y-q) - 13(z - r) = 0 ... plugging in coordinates from normal vector<BR>\n" );
document.write( "7(x-(-2)) + 17(y-3) - 13(z - 1) = 0 ... plugging in coordinates from point A in the plane<BR>\n" );
document.write( "7(x+2) + 17(y-3) - 13(z - 1) = 0<BR>\n" );
document.write( "7x+14 + 17y-51 - 13z + 13 = 0<BR>\n" );
document.write( "7x+17y-13z + 14-51+13 = 0<BR>\n" );
document.write( "7x+17y-13z - 24  = 0<BR>\n" );
document.write( "7x+17y-13z  = 24<BR>\n" );
document.write( "That is one possible answer for part (a).<BR>\n" );
document.write( "Other answers are possible because we could scale the equation up or down by some factor.\r<BR>\n" );
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document.write( "Explanation for part (b)\r<BR>\n" );
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document.write( "Let's find vector AB<BR>\n" );
document.write( "vector AB = B-A <BR>\n" );
document.write( "vector AB = < 3,-2,0 > - < 1,2,1 ><BR>\n" );
document.write( "vector AB = < 3-1,-2-2,0-1 ><BR>\n" );
document.write( "vector AB = < 2,-4,-1 ><BR>\n" );
document.write( "The direction vector of line AB is < 2,-4,-1 ><BR>\n" );
document.write( "Because this line is perpendicular to the plane we want, vector AB is a normal vector of this plane.\r<BR>\n" );
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document.write( "normal vector = < a,b,c > = < 2,-4,-1 ><BR>\n" );
document.write( "point on plane = (p,q,r) = (-1,1,0)\r<BR>\n" );
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document.write( "Equation of the plane in cartesian form<BR>\n" );
document.write( "a(x-p) + b(y-q) + c(z - r) = 0<BR>\n" );
document.write( "2(x - p) - 4(y-q) - 1(z - r) = 0  ... plugging in coordinates from normal vector<BR>\n" );
document.write( "2(x - (-1)) - 4(y - 1) - 1(z - 0) = 0  ... plugging in coordinates from point<BR>\n" );
document.write( "2(x+1) - 4(y-1) - z = 0<BR>\n" );
document.write( "2x + 2 - 4y + 4 - z = 0<BR>\n" );
document.write( "2x-4y-z+6 = 0<BR>\n" );
document.write( "2x+4y-z = -6\r<BR>\n" );
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document.write( "Further Reading<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.whitman.edu/mathematics/calculus_online/section12.05.html\">https://www.whitman.edu/mathematics/calculus_online/section12.05.html</a><BR>\n" );
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