The Slope of a Line

Inclination of a Line:
The angle from the X-axis to any given non horizontal line $$l$$ is called the inclination of line $$l$$. Here $$\alpha $$ is the inclination where $$0 < \alpha < {180^ \circ }$$, measured in a counter-clockwise direction from the positive X-axis to the line $$l$$.

Slope of a Line:
If $$\alpha $$ is the inclination of a non-vertical straight line $$l$$, then its slope or gradient is defined as $$\tan \alpha $$. The slope of a straight line is usually denoted by $$m$$, so the formula to find the slope of a line is given by
\[\boxed{m = \tan \alpha }\]

If a straight line is parallel to the X-axis, then its slope is equal to zero, i.e. $$m = \tan {0^ \circ } = 0$$. If the straight line is parallel to the Y-axis, then its slope is undefined, i.e. $$m = \tan {90^ \circ } = \infty $$.

If the line is parallel to the X-axis, then the ordinate of each point on the line is a fixed number, so its equation will be $$y = a$$, where $$a$$ is a fixed number. If the line is parallel to the Y-axis, then the abscissa of each point on the line is a fixed number, so its equation will be $$x = a$$, where $$a$$ is a fixed number.

Let $${m_1},{m_2}$$ be the slopes of the lines $${l_1},{l_2}$$ respectively.

(i) The lines $${l_1}$$ and $${l_2}$$ are parallel if and only if $${m_1} = {m_2}$$

(ii) The lines $${l_1}$$ and $${l_2}$$ are perpendicular if and only if $${m_1} \times {m_2} = – 1$$

Example: Find the slope of a straight line with inclination $${60^ \circ }$$ with the X-axis.
Here we have the inclination $$\alpha = {60^ \circ }$$ with the X-axis, now we shall find the slope of the straight line using the formula $$m = \tan \alpha $$,

\[m = \tan \alpha = \tan {60^ \circ } = \sqrt 3 \]