Coordinate Geometry

Perpendicular lines
Two lines or linear equations are perpendicular if their slopes are negative reciprocals of each other (opposite signs and upside down):

m1 = − 1/m2

Example: 5x − y = 8 and 5x = − x + 3

Let's check if these two functions are Perpendicular:
- first turn the fucntions into a slope intercept form (regular form of linear equations, in oder words solve for y): y = mx + b with m = slope

5x − y = 8 − y = 8 − 5x y = 5x − 8 The slope of this line is m1 = 5 or m1 = 5/1

5y = − x + 3 y = − 1/5·x − 3/5 The slope of this line is m2 = − 1/5

- then compare their slopes or gradients:
As you can notice m₁ = ⁵/₁ is exactly the negative reciprocal of m² = − ¹/₅ and vice versa.

⇒ Therefore, the lines 5x − y = 8 and 5x = − x + 3 are perpendicular!

Parallel lines
Two lines or linear equations are parallel if their slopes are equal (they have the same slopes). Since they'll never intersect, they continue forever without touching (assuming that these lines are on the same plane).

m1 = m2

Example:
One line passes through the points p₁(–1, –2) and p₂(1, 2); another line passes through the points q₁(–2, 0) and q₂(0, 4).

Our answer to this question is based on the calculation of slopes of both lines (functions).

⇒ With identical slopes, these lines are parallel!

Neither Parallel nor Perpendicular lines
Two lines or linear equations are neither parallel nor perpendicular if their slopes are neither the same and nor negative reciprocals of each other.

m1 ≠ m2 and m1 ≠ − 1/m2

Example:
A line passes through the points p₁(–4, 2) and p₂(0, 3); another line passes through the points q₁(–3, –2) and q₂(3, 2). Are these lines parallel, perpendicular, or neither?

These slope values are not the same, so the lines are not parallel. The slope values are not negative reciprocals either, so the lines are not perpendicular. ⇒ Then the answer is "neither".!

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