document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201855>Question 1201855</A>: <I><SPAN STYLE=\"word-break: break-all;\"> from my 'Vector Equations of Planes' lesson\r<BR>\n" );
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document.write( "the plane with the equation (vector) r = (1,2,3)+ s (1,2,5) + t(1,-1,3) where s, t E R inrersects the y and z-axes at the points A and B. Determine the equation of the line thru the points A and B.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836405 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=836405\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <BR>\n" );
document.write( "Either <BR>\n" );
document.write( "<font color=red size=4>< x,y,z > = < 0,3,0 > + t*< 0,-3,-2 ></font><BR>\n" );
document.write( "Or<BR>\n" );
document.write( "<font color=red size=4>< x,y,z > = < 0,3-3t,-2t ></font><BR>\n" );
document.write( "where t is any real number.\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "Each point in 3-space, aka 3d space, is of the form (x,y,z).\r<BR>\n" );
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document.write( "Points on the y axis have the x and z coordinates equal to zero. <BR>\n" );
document.write( "The y coordinate could be zero, or could be any other real number.<BR>\n" );
document.write( "So we have points of the form (0,y,0).<BR>\n" );
document.write( "Since point A is the y-intercept, let's have the point be labeled (0,A,0).\r<BR>\n" );
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document.write( "We also have (0,0,B) which represents the z intercept located at point B.\r<BR>\n" );
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document.write( "Now let's rewrite the equation of the plane.<BR>\n" );
document.write( "r = (1,2,3)+ s(1,2,5) + t(1,-1,3) <BR>\n" );
document.write( "r = (1,2,3)+ (s,2s,5s) + (t,-t,3t) <BR>\n" );
document.write( "r = (1+s+t,2+2s-t,3+5s+3t)<BR>\n" );
document.write( "(x,y,z) = (1+s+t,2+2s-t,3+5s+3t)\r<BR>\n" );
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document.write( "That last equation breaks down into these three equations<BR>\n" );
document.write( "x = 1+s+t<BR>\n" );
document.write( "y = 2+2s-t<BR>\n" );
document.write( "z = 3+5s+3t<BR>\n" );
document.write( "If we knew what s and t were, then we'd determine a location (x,y,z)<BR>\n" );
document.write( "Doing this a whole bunch of times for infinitely many r and s values will generate a flat plane that goes on forever in all directions.\r<BR>\n" );
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document.write( "Plug in the coordinates of the y intercept.<BR>\n" );
document.write( "0 = 1+s+t<BR>\n" );
document.write( "A = 2+2s-t<BR>\n" );
document.write( "0 = 3+5s+3t\r<BR>\n" );
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document.write( "Solve the subsystem<BR>\n" );
document.write( "0 = 1+s+t<BR>\n" );
document.write( "0 = 3+5s+3t<BR>\n" );
document.write( "and you should get the solution<BR>\n" );
document.write( "s = 0, t = -1<BR>\n" );
document.write( "I skipped the steps to solving, which I'll leave for the student to do. <BR>\n" );
document.write( "Let me know if you get stuck here.\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "A = 2+2s-t<BR>\n" );
document.write( "A = 2+2*0-(-1)<BR>\n" );
document.write( "A = 3<BR>\n" );
document.write( "Meaning the point A is located at (0,3,0) as the y intercept of this plane.\r<BR>\n" );
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document.write( "Now plug in the coordinates of the z intercept.<BR>\n" );
document.write( "0 = 1+s+t<BR>\n" );
document.write( "0 = 2+2s-t<BR>\n" );
document.write( "B = 3+5s+3t\r<BR>\n" );
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document.write( "Solve this subsystem<BR>\n" );
document.write( "0 = 1+s+t,<BR>\n" );
document.write( "0 = 2+2s-t<BR>\n" );
document.write( "to get<BR>\n" );
document.write( "s = -1, t = 0\r<BR>\n" );
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document.write( "Therefore,<BR>\n" );
document.write( "B = 3+5s+3t<BR>\n" );
document.write( "B = 3+5*(-1)+3*0<BR>\n" );
document.write( "B = -2<BR>\n" );
document.write( "The z intercept is located at (0,0,-2)\r<BR>\n" );
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document.write( "---------------------------\r<BR>\n" );
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document.write( "We now need to find the equation of the line through A(0,3,0) and B(0,0,-2)\r<BR>\n" );
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document.write( "Subtract the two sets of coordinates to determine the vector that points from A to B<BR>\n" );
document.write( "B-A = (0,0,-2) - (0,3,0)<BR>\n" );
document.write( "B-A = (0-0,0-3,-2-0)<BR>\n" );
document.write( "B-A = (0,-3,-2)<BR>\n" );
document.write( "To go from A to B, we do three things:<ul><li>Move 0 units along the x axis.</li><li>Move 3 units along the negative y axis.</li><li>Move 2 units along the negative z axis.</li></ul>This vector <0,-3,-2> gives us the direction in which to move from point to point along this line. <BR>\n" );
document.write( "It is analogous to the slope in 2D settings.\r<BR>\n" );
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document.write( "Let's say we started at A(0,3,0) which is the y intercept.<BR>\n" );
document.write( "Add on the direction vector mentioned earlier, but scaled up by some parameter t.\r<BR>\n" );
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document.write( "< x,y,z > = some vector in 3 space<BR>\n" );
document.write( "< x,y,z > = startPoint + t*(direction vector)<BR>\n" );
document.write( "<font color=red>< x,y,z > = < 0,3,0 > + t*< 0,-3,-2 ></font><BR>\n" );
document.write( "< x,y,z > = < 0,3,0 > + < t*0,t*(-3),t*(-2) ><BR>\n" );
document.write( "< x,y,z > = < 0,3,0 > + <0,-3t,-2t><BR>\n" );
document.write( "< x,y,z > = < 0+0,3+(-3t),0+(-2t) ><BR>\n" );
document.write( "< x,y,z > = < 0,3-3t,-2t ><BR>\n" );
document.write( "<font color=red>< x,y,z > = < 0,3-3t,-2t ></font> where t is any real number.\r<BR>\n" );
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document.write( "Let's see what happens when t = 0<BR>\n" );
document.write( "(x,y,z) = (0,3-3t,-2t)<BR>\n" );
document.write( "(x,y,z) = (0,3-3*0,-2*0)<BR>\n" );
document.write( "(x,y,z) = (0,3,0)<BR>\n" );
document.write( "We're at the point A(0,3,0) which is the y intercept.\r<BR>\n" );
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document.write( "Now try t = 1. <BR>\n" );
document.write( "This represents moving forward 1 unit in time.<BR>\n" );
document.write( "(x,y,z) = (0,3-3t,-2t)<BR>\n" );
document.write( "(x,y,z) = (0,3-3*1,-2*1)<BR>\n" );
document.write( "(x,y,z) = (0,0,-2)<BR>\n" );
document.write( "We've arrived at the z intercept (0,0,-2) which is point B.<BR>\n" );
document.write( "This confirms we have the correct equation of the line.<BR>\n" );
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