Parallel, perpendicular and collinear lines

Distinct nonvertical lines are parallel when their gradients are equal. Nonvertical perpendicular lines satisfy \(m_1m_2=-1\). A horizontal line and a vertical line are perpendicular; do not apply the product rule to an undefined gradient.

Example

A line perpendicular to \(y=2x+3\) has gradient \(-\dfrac12\). Through \((4,1)\), it satisfies \(y-1=-\dfrac12(x-4)\), hence \(y=-\dfrac12x+3\).

For three distinct points, equal successive gradients show collinearity when the gradients exist. If all three share an \(x\)-coordinate, they are collinear on a vertical line. Check distinctness before dividing coordinate differences.