An Introduction to the Vector Equation of a Line

Given two different points we can always draw a line segment between them.  We can also sketch a line given just one point and a direction.  How?  The instructions are ‘hiding in plain view’ in the equation y = mx + c. Take, for example y = ½ x – 3. 

Please do not make a table of values for x and y.

Tables are slow and inefficient.  Remember, we only need one point!  Which point? The y-intercept.  For the equation y = ½ x – 3, start at the point (0, c) = (0, -3).   Now what?  From the gradient, m = ½,  we find that the rise is 1 and the run is 2. 

If the gradient is written as a fraction (½, 23/7, -4/3, …) the numerator is the ‘rise’ and the denominator is the run.  If, however, the gradient is written as a decimal number (0.5, 1.6, - 3.9, …) the given number is the ‘rise’ and the run is always ‘1’. Positive runs go to the right; negative runs to the left.  Positive rises go up; negative rises go down. 

With preliminaries out of the way, here is the line: Start at the y-intercept, sketch the run (running parallel to the x-axis) and then sketch the rise (rising parallel to the y-axis).  Label the end point, P.   Take a ruler and connect the y-intercept and point P.

You can explore with the Geogebra file “mx_plus_c” to vary intercept c and gradient m using the respectively labeled sliders.  Turn on the trace function for point P by right clicking on the point and selecting “Trace On”.  Move the c and m sliders to choose your desired y intercept and gradient. After you have chosen these parameters, go the pull-down menu, View and click “Refresh Views”. Finally, go to the x_P slider and drag that button.  What happens?

The rise and run are continually refreshed, so the screen only shows the latest values, but we see all the previous values of point P.  The locus of point P is the required line. Ta Dah! 

Conclusion:

For many of you this essay merely re-stated the obvious.  However, the next installment will use very similar ideas to explain the vector equation of a line: 

p = q + kv .

Appendix:  Geogebra Construction

No.	Name		current value
1	Number, m	a slider variable	m = 1.7
2	Number, c	a slider variable	c = -1
3	Point, O	the origin	O = (0, 0)
4	Point, I	(0, c)	I = (0, -1)
5	Vector, vectorc	Vector[point, point] = Vector[O, I]	vectorc = (0, -1)
6	Number, xp	a slider variable	xp = 2.8
7	Vector run	Vector[point, point] = Vector[(0, c), (xp, c)]	run = (2.8, 0)
8	Vector rise	Vector[point, point] =Vector[(xp, c), (xp, c + xp times m)]	rise = (0, 4.76)
9	Point P	vectorc  + run + rise	P = (2.8, 3.76)

Created with GeoGebra

 A copy of the applet is stored on http://www.geogebratube.org/