document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1087143>Question 1087143</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hey I am asking for the proof of this formula<BR>\n" );
document.write( "a.b =|a|b|cos theta.\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #701667 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=701667\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You should revise your question.  \r<BR>\n" );
document.write( "\n" );
document.write( "——<BR>\n" );
document.write( "You have provided the geometric definition for the dot product (aka inner product) of two vectors.  The definition can be accepted straight away.    Are you asking to prove the algebraic definition of the dot product is equal to the geometric definition of the dot product?  \r<BR>\n" );
document.write( "\n" );
document.write( "Algebraic:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a_dot_b+=+a%5B1%5D%2Ab%5B1%5D+%2B+a%5B2%5D%2Ab%5B2%5D+%2B+a%5B3%5D%2Ab%5B3%5D+ ALIGN=MIDDLE  ALT=\"+a_dot_b+=+a%5B1%5D%2Ab%5B1%5D+%2B+a%5B2%5D%2Ab%5B2%5D+%2B+a%5B3%5D%2Ab%5B3%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" > + … + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%5Bn%5D%2Ab%5Bn%5D+ ALIGN=MIDDLE  ALT=\"+a%5Bn%5D%2Ab%5Bn%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Geometric:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a_dot_b+=+abs%28a%29abs%28b%29%2Acos%28theta%29+ ALIGN=MIDDLE  ALT=\"+a_dot_b+=+abs%28a%29abs%28b%29%2Acos%28theta%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > = angle between the two vectors\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "As I thought about it, I figured that must be what you want.  Here is the proof:<BR>\n" );
document.write( "I will use capital letters for vectors, lower case for scalers.   <BR>\n" );
document.write( "Also |A| means the magnitude of A, found using ordinary root-of-squares method.\r<BR>\n" );
document.write( "\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+E%5Bi%5D+ ALIGN=MIDDLE  ALT=\"+E%5Bi%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" > be a set of unit vectors in the n-dimensional space.<BR>\n" );
document.write( "So  <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+E%5Bi%5D%2AE%5Bj%5D+=+0++ ALIGN=MIDDLE  ALT=\"+E%5Bi%5D%2AE%5Bj%5D+=+0++\"   DISABLED_style= \"max-width:90%; height:auto;\" > if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++i%3C%3Ej+ ALIGN=MIDDLE  ALT=\"++i%3C%3Ej+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++E%5Bi%5D%2AE%5Bj%5D+=+1+ ALIGN=MIDDLE  ALT=\"++E%5Bi%5D%2AE%5Bj%5D+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+++i=j+%29+ ALIGN=MIDDLE  ALT=\"+++i=j+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+=+sum%28a%5Bi%5DE%5Bi%5D%2C+i=1%2C+n+%29+ ALIGN=MIDDLE  ALT=\"+A+=+sum%28a%5Bi%5DE%5Bi%5D%2C+i=1%2C+n+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >    (1)     and\r\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+B+=+sum%28b%5Bi%5DE%5Bi%5D%2C+i=1%2C+n%29+ ALIGN=MIDDLE  ALT=\"+B+=+sum%28b%5Bi%5DE%5Bi%5D%2C+i=1%2C+n%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >     (2)    <—<<<  Remember <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%5Bi%5D+ ALIGN=MIDDLE  ALT=\"+a%5Bi%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b%5Bi%5D+ ALIGN=MIDDLE  ALT=\"+b%5Bi%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" > are scalers \r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "\n" );
document.write( "and the algebraic dot product is defined as:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A%2AB+=+sum%28a%5Bi%5Db%5Bi%5D%2C+i=1%2C+n%29+ ALIGN=MIDDLE  ALT=\"+A%2AB+=+sum%28a%5Bi%5Db%5Bi%5D%2C+i=1%2C+n%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >     (*)<BR>\n" );
document.write( "—\r<BR>\n" );
document.write( "\n" );
document.write( "Now, use the geometric dot product definition to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A%2AE%5Bi%5D+ ALIGN=MIDDLE  ALT=\"+A%2AE%5Bi%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" > : <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A%2AE%5Bi%5D+=+abs%28A%29abs%28E%5Bi%5D%29cos%28theta%29+=+abs%28A%29%2A1%2Acos%28theta%29+=+a%5Bi%5D+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >  <—<<< Same <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%5Bi%5D+ ALIGN=MIDDLE  ALT=\"+a%5Bi%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" > as in (1)<BR>\n" );
document.write( "(and similarly for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+B%2AE%5Bi%5D+ ALIGN=MIDDLE  ALT=\"+B%2AE%5Bi%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" > )<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%5Bi%5D+ ALIGN=MIDDLE  ALT=\"+a%5Bi%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b%5Bi%5D+ ALIGN=MIDDLE  ALT=\"+b%5Bi%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be thought of as the magnitude of the component of A, B that project in the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E%5Bi%5D ALIGN=MIDDLE  ALT=\"E%5Bi%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > direction, respectively.     \r<BR>\n" );
document.write( "\n" );
document.write( "and substituting for B:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A%2AB+=+A%2Asum%28b%5Bi%5DE%5Bi%5D%2C+i=1%2C+n%29+ ALIGN=MIDDLE  ALT=\"+A%2AB+=+A%2Asum%28b%5Bi%5DE%5Bi%5D%2C+i=1%2C+n%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Finally, using the distributive property of dot products  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+C%2AkD+=+kC%2AD+ ALIGN=MIDDLE  ALT=\"+C%2AkD+=+kC%2AD+\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A%2AB+=+A%2Asum%28b%5Bi%5DE%5Bi%5D%2C+i=1%2C+n%29++=+sum%28b%5Bi%5DA%2AE%5Bi%5D%2C+i=1%2C+n%29++=+highlight%28sum%28b%5Bi%5Da%5Bi%5D%2C+i=1%2C+n%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >  <—<<< Same as (*) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );