How to Write Equations of Perpendicular & Parallel Lines

Parallel lines extend to infinity without crossing.

Parallel lines are straight lines that extend to infinity without touching at any point. Perpendicular lines cross each other at a 90-degree angle. Both sets of lines are important for many geometric proofs, so it is important to recognize them graphically and algebraically. You must know the structure of a straight-line equation before you can write equations for parallel or perpendicular lines. The standard form of the equation is "y = mx + b," in which "m" is the slope of the line and "b" is the point where the line crosses the y-axis.

Instructions

Parallel Lines

1. Write the equation for the first line and identify the slope and y-intercept.

   Example:
   y = 4x + 3
   m = slope = 4
   b = y-intercept = 3

2. Copy the first half of the equation for the parallel line. A line is parallel to another if their slopes are identical.

   Example:
   Original line: y = 4x + 3
   Parallel line: y = 4x

3. Choose a y-intercept different from the original line. Regardless of the magnitude of the new y-intercept, as long as the slope is identical, the two lines will be parallel.

   Example:
   Original line: y = 4x + 3
   Parallel line 1: y = 4x + 7
   Parallel line 2: y = 4x - 6
   Parallel line 3: y = 4x + 15,328.35

Perpendicular Lines

1. Write the equation for the first line and identify the slope and y-intercept, as with the parallel lines.

   Example:
   y = 4x + 3
   m = slope = 4
   b = y-intercept = 3

2. Transform for the "x" and "y" variable. The angle of rotation is 90 degrees because a perpendicular line intersects the original line at 90 degrees.

   Example:
   x' = x*cos(90) - y*sin(90)
   y' = x*sin(90) + y*cos(90)

   x' = -y
   y' = x

3. Substitute "y'" and "x'" for "x" and "y" and then write the equation in standard form.

   Example:
   Original line: y = 4x + 3
   Substitute: -x' = 4y' + 3
   Standard form: y' = -(1/4)*x - 3/4

   The original line, y = 4x + b, is perpendicular to new line, y' = -(1/4)*x - 3/4, and any line parallel to the new line, such as y' = -(1/4)*x - 10.

Tips & Warnings

• For three-dimensional lines, the process is the same but the calculations are much more complex. A study of Euler angles will help understand three-dimensional transformations.

References

• "Linear Algebra and its Applications"; Gilbert Strang; 1988