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font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html"><TR><TD><IMG height="1" width="400" alt="" src="http://mathworld.wolfram.com/images/spacer.gif" /></TD></TR><TR><TD align="left" width="100%" valign="top"><DIV class="subjectnavbar"> <A href="http://mathworld.wolfram.com/topics/Geometry.html">Geometry</A> &gt; <A href="http://mathworld.wolfram.com/topics/Distance.html">Distance</A> &gt; </DIV> <DIV class="subjectnavbar"> <A href="http://mathworld.wolfram.com/topics/Geometry.html">Geometry</A> &gt; <A href="http://mathworld.wolfram.com/topics/LineGeometry.html">Line Geometry</A> &gt; <A href="http://mathworld.wolfram.com/topics/Lines.html">Lines</A> &gt; </DIV> <TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD valign="baseline" class="title"><SPAN class="nowrap">Point-Line Distance--3-Dimensional</SPAN></TD></TR><TR><TD valign="top"><SPAN class="nowrap"><IMG height="3" width="100%" alt="" src="http://mathworld.wolfram.com/images/entries/underline.gif" /><IMG height="3" width="20" alt="" src="http://mathworld.wolfram.com/images/entries/underline.gif" /></SPAN></TD><TD><IMG height="15" alt="" src="http://mathworld.wolfram.com/images/spacer.gif" /></TD></TR></TBODY></TABLE></TD></TR><TR valign="top"><TD><A href="http://mathworld.wolfram.com/notebooks/PlaneGeometry/Point-LineDistance3-Dimensional.nb"><IMG height="26" border="0" width="119" alt="DOWNLOAD Mathematica Notebook" src="http://mathworld.wolfram.com/images/entries/dnld-nb.gif" /></A> <DIV align="center"> <IMG height="219" width="416" alt="PointLineDistance3D" src="http://mathworld.wolfram.com/images/eps-gif/PointLineDistance3D_1000.gif" /> </DIV> <P class="Text"> Let a line in three dimensions be specified by two points <IMG height="14" border="0" width="92" alt="x_1=(x_1,y_1,z_1)" class="inlineformula" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/Inline1.gif" /> and <IMG height="14" border="0" width="92" alt="x_2=(x_2,y_2,z_2)" class="inlineformula" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/Inline2.gif" /> lying on it, so a vector along the line is given by </p> <DIV> <TABLE cellspacing="0" cellpadding="0" align="center" width="100%"><TBODY><TR><TD align="left"><IMG height="56" border="0" width="128" alt=" v=[x_1+(x_2-x_1)t; y_1+(y_2-y_1)t; z_1+(z_2-z_1)t]. " class="numberedequation" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/NumberedEquation1.gif" /></TD><TD align="right" width="3"><DIV class="eqnum" id="eqn1"> (1) </DIV> </TD></TR></TBODY></TABLE></DIV> <P class="Text"> The squared distance between a point on the line with parameter <IMG height="14" border="0" width="4" alt="t" class="inlineformula" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/Inline3.gif" /> and a point <IMG height="16" border="0" width="92" alt="x_0=(x_0,y_0,z_0)" class="inlineformula" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/Inline4.gif" /> is therefore </p> <DIV> <TABLE cellspacing="0" cellpadding="0" align="center" width="100%"><TBODY><TR><TD align="left"><IMG height="19" border="0" width="471" alt=" d^2=[(x_1-x_0)+(x_2-x_1)t]^2+[(y_1-y_0)+(y_2-y_1)t]^2+[(z_1-z_0)+(z_2-z_1)t]^2. " class="numberedequation" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/NumberedEquation2.gif" /></TD><TD align="right" width="3"><DIV class="eqnum" id="eqn2"> (2) </DIV> </TD></TR></TBODY></TABLE></DIV> <P class="Text"> To minimize the distance, set <IMG height="21" border="0" width="79" alt="d(d^2)/dt=0" class="inlineformula" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/Inline5.gif" /> and solve for <IMG height="14" border="0" width="4" alt="t" class="inlineformula" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/Inline6.gif" /> to obtain </p> <DIV> <TABLE cellspacing="0" cellpadding="0" align="center" width="100%"><TBODY><TR><TD align="left"><IMG height="39" border="0" width="145" alt=" t=-((x_1-x_0)·(x_2-x_1))/(|x_2-x_1|^2), " class="numberedequation" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/NumberedEquation3.gif" /></TD><TD align="right" width="3"><DIV class="eqnum" id="eqn3"> (3) </DIV> </TD></TR></TBODY></TABLE></DIV> <P class="Text"> where <IMG height="14" border="0" width="3" alt="·" class="inlineformula" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/Inline7.gif" /> denotes the <A class="Hyperlink" href="http://mathworld.wolfram.com/DotProduct.html">dot product</A>. The minimum distance can then be found by plugging <IMG height="14" border="0" width="4" alt="t" class="inlineformula" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/Inline8.gif" /> back into (<A class="Hyperlink" href="#eqn2">2</A>) to obtain </p> <TABLE cellspacing="0" cellpadding="0" border="0" align="center" width="100%"><TBODY><TR><TD align="right" width="1"><IMG height="21" border="0" width="828" alt="d^2=(x_1-x_0)^2+(y_1-y_0)^2+(z_1-z_0)^2+2t[(x_2-x_1)(x_1-x_0)+(y_2-y_1)(y_1-y_0)+(z_2-z_1)(z_1-z_0)]+t^2[(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2] " class="displayformula" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/Inline9.gif" /></TD><TD align="right" width="10"><DIV class="eqnum" id="eqn4"> (4) </DIV> </TD></TR><TR><TD align="right" width="1"><IMG height="43" border="0" width="366" alt="=|x_1-x_0|^2-2([(x_1-x_0)·(x_2-x_1)]^2)/(|x_2-x_1|^2)+([(x_1-x_0)·(x_2-x_1)]^2)/(|x_2-x_1|^2) " class="displayformula" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/Inline10.gif" /></TD><TD align="right" width="10"><DIV class="eqnum" id="eqn5"> (5) </DIV> </TD></TR><TR><TD align="right" width="1"><IMG height="43" border="0" width="276" alt="=(|x_1-x_0|^2|x_2-x_1|^2-[(x_1-x_0)·(x_2-x_1)]^2)/(|x_2-x_1|^2). " class="displayformula" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/Inline11.gif" /></TD><TD align="right" width="10"><DIV class="eqnum" id="eqn6"> (6) </DIV> </TD></TR></TBODY></TABLE><P class="Text"> Using the <A class="Hyperlink" href="http://mathworld.wolfram.com/VectorQuadrupleProduct.html">vector quadruple product</A> </p> <DIV> <TABLE cellspacing="0" cellpadding="0" align="center" width="100%"><TBODY><TR><TD align="left"><IMG height="17" border="0" width="148" alt=" (AxB)^2=A^2B^2-(A·B)^2 " class="numberedequation" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/NumberedEquation4.gif" /></TD><TD align="right" width="3"><DIV class="eqnum" id="eqn7"> (7) </DIV> </TD></TR></TBODY></TABLE></DIV> <P class="Text"> where <IMG height="14" border="0" width="7" alt="x" class="inlineformula" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/Inline12.gif" /> denotes the <A class="Hyperlink" href="http://mathworld.wolfram.com/CrossProduct.html">cross product</A> then gives </p> <DIV> <TABLE cellspacing="0" cellpadding="0" align="center" width="100%"><TBODY><TR><TD align="left"><IMG height="43" border="0" width="158" alt=" d^2=(|(x_2-x_1)x(x_1-x_0)|^2)/(|x_2-x_1|^2), " class="numberedequation" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/NumberedEquation5.gif" /></TD><TD align="right" width="3"><DIV class="eqnum" id="eqn8"> (8) </DIV> </TD></TR></TBODY></TABLE></DIV> <P class="Text"> and taking the square root results in the beautiful formula </p> <DIV> <TABLE cellspacing="0" cellpadding="0" align="center" width="100%"><TBODY><TR><TD align="left"><IMG height="38" border="0" width="146" alt=" d=(|(x_2-x_1)x(x_1-x_0)|)/(|x_2-x_1|). " class="numberedequation" src="http://mathworld.wolfram.com/images/equations/Point-LineDistance3-Dimensional/NumberedEquation6.gif" /></TD><TD align="right" width="3"><DIV class="eqnum" id="eqn9"> (9) </DIV> </TD></TR></TBODY></TABLE></DIV> <FORM action="http://mathworld.wolfram.com/search/" method="get" name="SearchLinks"> <P class="CrossRefs"> <SPAN class="crosslinkheader">SEE ALSO:</SPAN> <A class="Hyperlink" href="http://mathworld.wolfram.com/Collinear.html">Collinear</A>, <A class="Hyperlink" href="http://mathworld.wolfram.com/Line.html">Line</A>, <A class="Hyperlink" href="http://mathworld.wolfram.com/Point.html">Point</A>, <A class="Hyperlink" href="http://mathworld.wolfram.com/Point-LineDistance2-Dimensional.html">Point-Line Distance--2-Dimensional</A> </p> </FORM> <DIV> <IMG height="3" width="300" alt="" src="http://mathworld.wolfram.com/images/entries/underline.gif" /><BR /><BR /><BR /><SPAN class="crosslinkheader">CITE THIS AS:</SPAN> <P class="citation"> <A href="http://mathworld.wolfram.com/about/author.html">Weisstein, Eric W.</A> "Point-Line Distance--3-Dimensional." From <A href="http://mathworld.wolfram.com/"><I>MathWorld</I></A>--A Wolfram Web Resource. <A href="http://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html">http://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html</A></P></DIV></TD></TR></blockquote></div><div style="margin: 0px 6px 6px 4px;"><table style="font-size: 11px;border-spacing: 0px;padding: 0px;" cellpadding="0" cellspacing="0" width="100%"><tr><td style="background:transparent;border-width:0px;padding:0px;">&nbsp;</td><td align="right" style="background:transparent;border-width:0px;padding:0px;width:107px" width="107"><a href="http://clipmarks.com/share/49676482-E8C8-4EAE-B3E2-56C249F3908B/blog/" title="blog or email this clip"><img src="http://content6.clipmarks.com/images/c2b-foot.png" border="0" alt="blog it" width="107" height="17" style="border-width:0px;padding:0px;margin:0px;" /></a></td></tr></table></div></div>

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