document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207841>Question 1207841</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine whether each pair of lines is parallel, perpendicular, or neither. \r<BR>\n" );
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document.write( "(a) 3x + 4y = 12; 4x - 3y = -12 \r<BR>\n" );
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document.write( "(b) y = 1; y = -1 </SPAN>\n" );
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document.write( "For the first one, the other tutor uses a basic algebraic method, showing the lines are perpendicular by finding the slopes and showing that the product of the slopes is -1.<br><BR>\n" );
document.write( "A more sophisticated (and faster and easier) method is to use a dot product.<br><BR>\n" );
document.write( "Given two linear equations ax+by=m and cx+dy=n, the dot product is ac+bd -- the product of the x coefficients plus the product of the y coefficients.  The lines are perpendicular only if the dot product is 0.<br><BR>\n" );
document.write( "In the first example, the dot product is (3)(4)+(4)(-3) = 12-12 = 0, so the lines are perpendicular.<br><BR>\n" );
document.write( "With a little experience, you can tell that the lines are perpendicular by inspection.  By comparing the two equations, we see that the coefficients of x and y are switched, with one of them changing sign.  That guarantees that the dot product will be 0 and the lines will be perpendicular.<br><BR>\n" );
document.write( "Here are a couple of quick examples of pair of equations of lines that are perpendicular:<BR>\n" );
document.write( "3x-7y=4 and 7x+3y=4<BR>\n" );
document.write( "17x+39y=100 and 39x-17y=0<br><BR>\n" );
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