Parallel and perpendicular lines

Parallel lines have equal gradients

• Parallel lines run in the same direction and never meet — so they have the same steepness.
• Two lines are parallel when their gradients are equal:

$m_1 = m_2$

• To test if lines are parallel, put both in the form $y = mx + c$ and compare the gradients. Different intercepts, same gradient ⇒ parallel.

Perpendicular lines have gradients that multiply to −1

• Perpendicular lines cross at a right angle ($90^\circ$).
• Their gradients multiply to give $-1$:

$m_1 \times m_2 = -1 \qquad \text{equivalently} \qquad m_2 = -\frac{1}{m_1}$

• In words: the perpendicular gradient is the negative reciprocal — flip the fraction and change the sign.
  • gradient $2 \Rightarrow$ perpendicular gradient $-\tfrac{1}{2}$
  • gradient $-\tfrac{3}{4} \Rightarrow$ perpendicular gradient $\tfrac{4}{3}$

Finding the equation of a parallel or perpendicular line

• Read the gradient of the given line (rearrange to $y = mx + c$ first).
• Keep the gradient for a parallel line; take the negative reciprocal for a perpendicular line.
• Use the new gradient with the point you are given in $y - y_1 = m(x - x_1)$.