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-line

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