Gradients of intervals and lines

The gradient formula, parallel and perpendicular gradients, testing for collinear points, and the angle of inclination m = tan θ.

Gradient

Gradient formula

$$m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$$

Lines going up to the right have positive gradient; lines going down have negative gradient. Horizontal lines have gradient $0$, and vertical lines have no gradient (it is undefined).

The angle of inclination

The angle of inclination $\theta$ is the angle a line makes with the positive direction of the $x$-axis, measured anticlockwise, with $0^\circ \le \theta < 180^\circ$.

Gradient and angle

$$m = \tan\theta$$

Gradient from the angle

Acute angles give positive gradients; obtuse angles give negative gradients.

Try this

Find the angle that gives gradient $-1$.

Example 1●

Find the angle of inclination of the line through $(0, 1)$ and $(3, 7)$, to the nearest degree.

$m = \dfrac{7 - 1}{3 - 0} = 2$

$\tan\theta = 2$, so $\theta \approx 63^\circ$.

Example 2●●

A line has angle of inclination $150^\circ$. Find its exact gradient.

$m = \tan 150^\circ = -\tan 30^\circ = -\dfrac{1}{\sqrt{3}}$

150° is in quadrant 2, where tan is negative; related angle 30°.

Parallel and perpendicular lines

Perpendicular gradients

Orange: gradient m. Blue: gradient −1/m. They always meet at a right angle.

Try this

Why does this fail when $m = 0$?

Example 3: collinear points●●

Are $A(1, 2)$, $B(3, 6)$ and $C(6, 12)$ collinear?

$m_{AB} = \dfrac{6 - 2}{3 - 1} = 2$ and $m_{BC} = \dfrac{12 - 6}{6 - 3} = 2$

Yes, they are collinear.

Same gradient and a shared point B, so they are on one line.

Flashcards

Gradient formula

$\dfrac{y_2 - y_1}{x_2 - x_1}$

Parallel lines

$m_1 = m_2$

Perpendicular lines

$m_1 m_2 = -1$

Angle of inclination

$m = \tan\theta$

Collinear

Equal gradients through a common point.

Practice

Find the gradient of the line through $(-2, 5)$ and $(4, 2)$.
A line perpendicular to one with gradient $4$ has gradient:
A line has gradient $1$. Find its angle of inclination.
A line has angle of inclination $135^\circ$. Its gradient is:
Which pair of gradients belongs to perpendicular lines?
Are $(0, 1)$, $(2, 4)$ and $(4, 7)$ collinear?
The line through $(1, 2)$ and $(3, k)$ is parallel to $y = 2x + 5$. Find $k$.
Find the angle of inclination of a line with gradient $-\tfrac12$, to the nearest degree.