document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201857>Question 1201857</A>: <I><SPAN STYLE=\"word-break: break-all;\"> from my 'Vector Equation of a Line' lesson\r<BR>\n" );
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document.write( "A line has the equation y = (-5/6)x + 9\r<BR>\n" );
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document.write( "a. Give a direction vector for a line that is parallel to this line. \r<BR>\n" );
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document.write( "b. Give a direction vector for a line that is perpendicular to this line. \r<BR>\n" );
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document.write( "c. Give the coordinates of a point on the given line. \r<BR>\n" );
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document.write( "d. In both vector and parametric form, give the equations of the line parallel to the given line and passing through A (7,9)\r<BR>\n" );
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document.write( "e. In both vector and parametric form, give the equations of the line perpendicular to the given line and passing through B (-2,1) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836450 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=836450\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
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document.write( "The line is of the form y = mx+b<BR>\n" );
document.write( "m = slope = -5/6<BR>\n" );
document.write( "b = y intercept = 9\r<BR>\n" );
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document.write( "The slope tells us how to go from one point to another on the line.<BR>\n" );
document.write( "It is the direction vector.\r<BR>\n" );
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document.write( "slope = rise/run<BR>\n" );
document.write( "rise/run = -5/6<BR>\n" );
document.write( "rise = -5<BR>\n" );
document.write( "run = 6\r<BR>\n" );
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document.write( "The rise tells us how much to move up or down.<BR>\n" );
document.write( "In this case we go down 5. This is the change in y.<BR>\n" );
document.write( "The run is the change in x. We go 6 units to the right.\r<BR>\n" );
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document.write( "Answer: <font color=red size=4> < 6,-5 > </font> \r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Swap the coordinate positions of the direction vector. <BR>\n" );
document.write( "Then flip the sign of exactly one coordinate. \r<BR>\n" );
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document.write( "original = < 6,-5 ><BR>\n" );
document.write( "swapped = < -5,6 ><BR>\n" );
document.write( "change sign of x coord = < 5,6 ><BR>\n" );
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document.write( "change sign of y coord = < -5,-6 >\r<BR>\n" );
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document.write( "You can use the dot product to confirm vectors < 6,-5 > and < 5,6 > are perpendicular (same goes for < 6,-5 > and < -5,-6 > being perpendicular).<BR>\n" );
document.write( "If u dot v = 0, then u is perpendicular to v. \r<BR>\n" );
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document.write( "Answer: <font color=red size=4> < 5,6 > </font>  or <font color=red size=4> < -5,-6 > </font> \r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "Plug in x = 0 and find y<BR>\n" );
document.write( "y = (-5/6)*x + 9<BR>\n" );
document.write( "y = (-5/6)*0 + 9<BR>\n" );
document.write( "y = 0 + 9<BR>\n" );
document.write( "y = 9<BR>\n" );
document.write( "The point (0,9) is on this line\r<BR>\n" );
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document.write( "Repeat for x = 6<BR>\n" );
document.write( "y = (-5/6)*x + 9<BR>\n" );
document.write( "y = (-5/6)*6 + 9<BR>\n" );
document.write( "y = -5 + 9<BR>\n" );
document.write( "y = 4<BR>\n" );
document.write( "The point (6,4) is also on the line\r<BR>\n" );
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document.write( "The movement from (0,9) to (6,4) is \"down 5, right 6\"<BR>\n" );
document.write( "Or along the direction vector < 6,-5 > which means \"go right 6, then down 5\".<BR>\n" );
document.write( "There are infinitely many points on this line.\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>(0,9)</font> \r<BR>\n" );
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document.write( "Part (d)\r<BR>\n" );
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document.write( "Vector form of the line:<BR>\n" );
document.write( "< x,y > = startVector + t*DirectionVector<BR>\n" );
document.write( "< x,y > = < 0,9 > + t*< 6,-5 ><BR>\n" );
document.write( "The start vector could be any other vector you want, as long as it's on the line. <BR>\n" );
document.write( "So you could pick < 6,4 > for instance.\r<BR>\n" );
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document.write( "Now we need to find the vector equation of a line parallel to what was mentioned, but goes through (7,9)<BR>\n" );
document.write( "The start vector will be < 7,9 ><BR>\n" );
document.write( "The direction vector is the same. <BR>\n" );
document.write( "Parallel vectors are equal or scalar multiples of one another. They point in the same direction (eg: northeast).\r<BR>\n" );
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document.write( "Vector form of the parallel line:<BR>\n" );
document.write( "< x,y > = startVector + t*DirectionVector<BR>\n" );
document.write( "< x,y > = < 7,9 > + t*< 6,-5 >\r<BR>\n" );
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document.write( "Now rewrite things a bit like so<BR>\n" );
document.write( "< x,y > = < 7,9 > + t*< 6,-5 ><BR>\n" );
document.write( "< x,y > = < 7,9 > + < 6t,-5t ><BR>\n" );
document.write( "< x,y > = < 7+6t,9-5t ><BR>\n" );
document.write( "That breaks down into<BR>\n" );
document.write( "x = 7+6t<BR>\n" );
document.write( "y = 9-5t<BR>\n" );
document.write( "Both of which form a system of equations to define the parametric form of the parallel line.<BR>\n" );
document.write( "The t is any real number. It can be thought of as the time value.\r<BR>\n" );
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document.write( "What happens at t = 0?<BR>\n" );
document.write( "(x,y) = (7+6t,9-5t)<BR>\n" );
document.write( "(x,y) = (7+6*0,9-5*0)<BR>\n" );
document.write( "(x,y) = (7,9)<BR>\n" );
document.write( "Which confirms (7,9) is on the parallel line\r<BR>\n" );
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document.write( "Answers: <BR>\n" );
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document.write( "<tr><TD >Vector form:</TD><TD ><font color=red size=4>< x,y > = < 7,9 > + t*< 6,-5 > </font> </TD></tr><tr><TD >Parametric form:</TD><TD ><font color=red size=4>x = 7+6t</br>y = 9-5t</font></TD></tr>\n" );
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document.write( "Part (e)\r<BR>\n" );
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document.write( "Original direction vector = < 6,-5 ><BR>\n" );
document.write( "Perpendicular direction vector = < 5,6 ><BR>\n" );
document.write( "Refer to part (b)\r<BR>\n" );
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document.write( "Vector form of the perpendicular line:<BR>\n" );
document.write( "< x,y > = startVector + t*DirectionVector<BR>\n" );
document.write( "< x,y > = < -2,1 > + t*< 5,6 ><BR>\n" );
document.write( "< x,y > = < -2,1 > + < 5t,6t ><BR>\n" );
document.write( "< x,y > = < -2+5t,1+6t >\r<BR>\n" );
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document.write( "Any point on this perpendicular line through (-2,1) is of the general form (x,y) = (-2+5t, 1+6t) where t is any real number. <BR>\n" );
document.write( "Plug t = 0 to find (x,y) = (-2,1)\r<BR>\n" );
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document.write( "Answers: <BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Vector form:</TD><TD ><font color=red size=4>< x,y > = < -2,1 > + t*< 5,6 > </font> </TD></tr><tr><TD >Parametric form:</TD><TD ><font color=red size=4>x = -2+5t</br>y = 1+6t</font></TD></tr>\n" );
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