document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=833927>Question 833927</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine whether the three points P(-6, -9, 4), Q(-8, -13, -2) and R(-10, -17, -10) are collinear by computing the distances between pairs of points.  \r<BR>\n" );
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document.write( "(a) Distance from P to Q: <BR>\n" );
document.write( "(b) Distance from Q to R:  <BR>\n" );
document.write( "(c) Distance from P to R:  <BR>\n" );
document.write( "(d) Are the three points colinear (y/n)?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #502846 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=502846\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is one ugly problem.<BR>\n" );
document.write( "Calculating distances is a cumbersome way to find if the points are colinear.<BR>\n" );
document.write( "The answer for part (d) could be more easily found in different ways.<BR>\n" );
document.write( "I can see they are not colinear by simple mental math, but to answer parts (a), (b), and (c) we need to calculate those distances.<BR>\n" );
document.write( "Maybe the problem was designed to make students practice concentration, arithmetic and patience,<BR>\n" );
document.write( "or maybe students were taught to use a fancy calculator/computer programs to calculate distances without having to think.<BR>\n" );
document.write( "I don't like it. I like thinking, understanding, and solving problems by the shortest, more direct route.<BR>\n" );
document.write( "However, no one else seemed to want to solve this problem, so I might as well do it.<BR>\n" );
document.write( " <BR>\n" );
document.write( "If P, Q and R are colinear, two of the distance will add up to the third distance.<BR>\n" );
document.write( "If not, the points will be forming a triangle and the sum of the two shortest distances will be greater than the longer distance.<BR>\n" );
document.write( " <BR>\n" );
document.write( "CALCULATING DISTANCES:<BR>\n" );
document.write( "The distance between two points is<BR>\n" );
document.write( "the square root of<BR>\n" );
document.write( "the sum of the squares of<BR>\n" );
document.write( "the differences in the points coordinates.<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE: That is an ugly mouthful that correspond to the uglier formula below,<BR>\n" );
document.write( "but the concept is simple and neat;<BR>\n" );
document.write( "just apply the Pythagorean theorem once if given two variables,<BR>\n" );
document.write( "twice if given 3 variables.<BR>\n" );
document.write( "If this NOTE makes sense to you, enjoy it. If it does not make sense, ignore it.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The distance between points <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%5BP%5D%2Cy%5BP%5D%2Cz%5BP%5D%29 ALIGN=MIDDLE  ALT=\"P%28x%5BP%5D%2Cy%5BP%5D%2Cz%5BP%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Q%28x%5BQ%5D%2Cy%5BQ%5D%2Cz%5BQ%5D%29 ALIGN=MIDDLE  ALT=\"Q%28x%5BQ%5D%2Cy%5BQ%5D%2Cz%5BQ%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ=sqrt%28%28x%5BQ%5D-x%5BP%5D%29%5E2%2B%28y%5BQ%5D-y%5BP%5D%29%5E2%2B%28z%5BQ%5D-z%5BP%5D%29%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ=sqrt%28%28-8-%28-6%29%29%5E2%2B%28-13-%28-9%29%29%5E2%2B%28-2-4%29%5E2%29 ALIGN=MIDDLE  ALT=\"PQ=sqrt%28%28-8-%28-6%29%29%5E2%2B%28-13-%28-9%29%29%5E2%2B%28-2-4%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ=sqrt%28%28-8%2B6%29%5E2%2B%28-13%2B9%29%5E2%2B%28-6%29%5E2%29 ALIGN=MIDDLE  ALT=\"PQ=sqrt%28%28-8%2B6%29%5E2%2B%28-13%2B9%29%5E2%2B%28-6%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ=sqrt%28%28-2%29%5E2%2B%28-4%29%5E2%2B%28-6%29%5E2%29 ALIGN=MIDDLE  ALT=\"PQ=sqrt%28%28-2%29%5E2%2B%28-4%29%5E2%2B%28-6%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ=sqrt%284%2B16%2B36%29 ALIGN=MIDDLE  ALT=\"PQ=sqrt%284%2B16%2B36%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ=sqrt%2856%29=sqrt%284%2A14%29=2sqrt%2814%29=%22approximately+7.48%22 ALIGN=MIDDLE  ALT=\"PQ=sqrt%2856%29=sqrt%284%2A14%29=2sqrt%2814%29=%22approximately+7.48%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Similarly,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PR=sqrt%28%28-10-%28-6%29%29%5E2%2B%28-17-%28-9%29%29%5E2%2B%28-10-4%29%5E2%29 ALIGN=MIDDLE  ALT=\"PR=sqrt%28%28-10-%28-6%29%29%5E2%2B%28-17-%28-9%29%29%5E2%2B%28-10-4%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PR=sqrt%28%28-10%2B6%29%5E2%2B%28-17%2B9%29%5E2%2B%28-10-4%29%5E2%29 ALIGN=MIDDLE  ALT=\"PR=sqrt%28%28-10%2B6%29%5E2%2B%28-17%2B9%29%5E2%2B%28-10-4%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PR=sqrt%28%28-4%29%5E2%2B%28-8%29%5E2%2B%28-14%29%5E2%29 ALIGN=MIDDLE  ALT=\"PR=sqrt%28%28-4%29%5E2%2B%28-8%29%5E2%2B%28-14%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PR=sqrt%2816%2B64%2B196%29 ALIGN=MIDDLE  ALT=\"PR=sqrt%2816%2B64%2B196%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PR=sqrt%28276%29=sqrt%284%2A69%29=2sqrt%2869%29=%22approximately+16.61%22 ALIGN=MIDDLE  ALT=\"PR=sqrt%28276%29=sqrt%284%2A69%29=2sqrt%2869%29=%22approximately+16.61%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=RQ=sqrt%28%28-8-%28-10%29%29%5E2%2B%28-13-%28-17%29%29%5E2%2B%28-2-%28-10%29%29%5E2%29 ALIGN=MIDDLE  ALT=\"RQ=sqrt%28%28-8-%28-10%29%29%5E2%2B%28-13-%28-17%29%29%5E2%2B%28-2-%28-10%29%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=RQ=sqrt%28%28-8%2B10%29%5E2%2B%28-13%2B17%29%5E2%2B%28-2%2B10%29%5E2%29 ALIGN=MIDDLE  ALT=\"RQ=sqrt%28%28-8%2B10%29%5E2%2B%28-13%2B17%29%5E2%2B%28-2%2B10%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=RQ=sqrt%282%5E2%2B4%5E2%2B8%5E2%29 ALIGN=MIDDLE  ALT=\"RQ=sqrt%282%5E2%2B4%5E2%2B8%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=RQ=sqrt%284%2B16%2B64%29 ALIGN=MIDDLE  ALT=\"RQ=sqrt%284%2B16%2B64%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=RQ=sqrt%2884%29=sqrt%284%2A21%29=2sqrt%2821%29=%22approximately+9.17%22 ALIGN=MIDDLE  ALT=\"RQ=sqrt%2884%29=sqrt%284%2A21%29=2sqrt%2821%29=%22approximately+9.17%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I do not see any benefit from working with those pesky square roots, so let's add the approximate values. They are correct to within <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.01 ALIGN=MIDDLE  ALT=\"0.01\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "so assuming the sum of the shortest distance added to exactly the longest distance,<BR>\n" );
document.write( "we could not be off by more than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.01%2B0.01%2B0.01=0.03 ALIGN=MIDDLE  ALT=\"0.01%2B0.01%2B0.01=0.03\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7.48%2B9.17=16.65 ALIGN=MIDDLE  ALT=\"7.48%2B9.17=16.65\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the sum,<BR>\n" );
document.write( "and it differs from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16.61 ALIGN=MIDDLE  ALT=\"16.61\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16.65-16.61=0.04%3E0.03 ALIGN=MIDDLE  ALT=\"16.65-16.61=0.04%3E0.03\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so the points are not colinear. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );