Straight-line explorer

The gradient measures the direction of a line, so comparing gradients tells you at once whether two lines are parallel, perpendicular, or neither. Parallel: m_1=m_2 — the lines make equal angles with the x-axis and never meet. Perpendicular: m_1m_2=-1 — the lines cross at right angles. **Perpendicular gradient: change the sign and flip the fraction** **Gradient of l_1** | **Gradient of the perpendicular line** | **Check** 34 | - 43 | 34 x (- 43)=-1 - 25 | 52 | - 25 x 52=-1 6 | - 16 | 6 x (- 16)=-1 The perpendicular gradient is the negative reciprocal - 1m, not -m. Changing only the sign gives a line of equal steepness sloping the other way — not a perpendicular line.

The gradient measures the direction of a line, so comparing gradients tells you at once whether two lines are parallel, perpendicular, or neither.

**Perpendicular gradient: change the sign and flip the fraction**

The perpendicular gradient is the negative reciprocal \(-\dfrac{1}{m}\), not \(-m\). Changing only the sign gives a line of equal steepness sloping the other way — not a perpendicular line.

Gradient of \(l_1\)	Gradient of the perpendicular line	Check
\(\dfrac{3}{4}\)	\(-\dfrac{4}{3}\)	\(\dfrac{3}{4}\times\left(-\dfrac{4}{3}\right)=-1\)
\(-\dfrac{2}{5}\)	\(\dfrac{5}{2}\)	\(-\dfrac{2}{5}\times\dfrac{5}{2}=-1\)
\(6\)	\(-\dfrac{1}{6}\)	\(6\times\left(-\dfrac{1}{6}\right)=-1\)

Parallel and Perpendicular Lines

Parallel and Perpendicular Lines