document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201854>Question 1201854</A>: <I><SPAN STYLE=\"word-break: break-all;\"> from \"Vector Equations of Planes\" lesson.\r<BR>\n" );
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document.write( "A plane passes through the points P(-2,3,1), Q(-2,3,2) and R (1,0,1)\r<BR>\n" );
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document.write( "a. using vectors PQ and vectors PR as your direction vectors, write the vector equation of this plane\r<BR>\n" );
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document.write( "b. using vector QR, and any other direction vector, write a second vector equation for this plane.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836429 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=836429\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
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document.write( "Answer: <font color=red>< x,y,z > = < -2,3,1 > + s*< 0,0,1 > + t*< 3,-3,0 ></font><BR>\n" );
document.write( "where s and t are any real numbers\r<BR>\n" );
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document.write( "Explanation:<BR>\n" );
document.write( "The < -2,3,1 > is the position vector of point P. You can replace this position vector with the coordinates of Q or R.<BR>\n" );
document.write( "You can pick any point in the plane.<BR>\n" );
document.write( "This start position can be somewhat analogous to the y intercept<BR>\n" );
document.write( "y = mx+b has m = slope and b = y intercept<BR>\n" );
document.write( "b = start position<BR>\n" );
document.write( "m = tells us how to move = direction vector<BR>\n" );
document.write( "That's one way we can connect the ideas of 2D graphs with a 3D one like this.\r<BR>\n" );
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document.write( "The < 0,0,1 > represents the coordinates of vector PQ. It starts at P and points to Q.<BR>\n" );
document.write( "Subtract corresponding coordinates to determine this vector<BR>\n" );
document.write( "vector PQ = Q - P = < -2,3,2 > - < -2,3,1 > = < 0,0,1 ><BR>\n" );
document.write( "The < 3,-3,0 > refers to vector PR which is calculated in a similar fashion.\r<BR>\n" );
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document.write( "The template can be written as such<BR>\n" );
document.write( "< x,y,z > = positionVector + s*DirectionVector1 + t*DirectionVector2<BR>\n" );
document.write( "and more specifically as this template<BR>\n" );
document.write( "< x,y,z > = P + s*VectorPQ + t*VectorPR\r<BR>\n" );
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document.write( "We need 2 direction vectors because the plane is 2 dimensional. <BR>\n" );
document.write( "We have 2 degrees of freedom of where to go along the flat surface. <BR>\n" );
document.write( "Think of an xy axis. <BR>\n" );
document.write( "The two direction vectors cannot lie on the same line. <BR>\n" );
document.write( "Otherwise, infinitely many planes will result.\r<BR>\n" );
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document.write( "How can we determine if two vectors are on the same line or not?<BR>\n" );
document.write( "By solving for k in this vector equation<BR>\n" );
document.write( "PQ = k*PR<BR>\n" );
document.write( "< 0,0,1 > = k*< 3,-3,0 ><BR>\n" );
document.write( "< 0,0,1 > = < 3k, -3k, 0 ><BR>\n" );
document.write( "It should be fairly clear that there aren't any solutions for k. <BR>\n" );
document.write( "The last entries of 1 and 0 don't match up no matter what k would be.<BR>\n" );
document.write( "Therefore, there is no way to scale PR to get PQ, and vice versa.<BR>\n" );
document.write( "Furthermore, vectors PQ and PR are not on the same line.<BR>\n" );
document.write( "This allows us to use them as direction vectors of the plane.\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Answer:  <font color=red>< x,y,z > = < -2,3,1 > + s*< 0,0,-1> + t*< 3,-3,-1 ></font><BR>\n" );
document.write( "where s and t are any real numbers\r<BR>\n" );
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document.write( "Explanation:<BR>\n" );
document.write( "Start with the answer from part (a)<BR>\n" );
document.write( "Flip the signs of vector PQ to get vector QP, so we get < 0,0,-1> as another possible direction vector.<BR>\n" );
document.write( "Replace < 3,-3,0 >, which was from vector PR, with < 3,-3,-1 > which is vector QR.<BR>\n" );
document.write( "The calculation of vector QR is similar to what is shown in part (a) when we found vector PQ.<BR>\n" );
document.write( "The position vector can stay the same. <BR>\n" );
document.write( "Although as mentioned earlier, you can replace the position vector with the coordinates from Q or R.\r<BR>\n" );
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document.write( "I'll let you check if vectors QP and QR are on the same straight line or not.<BR>\n" );
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