Inclination of a line

Mathematics Class XI Chapter 1: Straight Lines Inclination of a line

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Inclination of a line is a real number $\theta$ as defined below:

(i) If a line $L$ is parallel to $x$ – axis or coincides with $x$ -axis then $\theta =0$

$L \parallel x- \text {axis}$

L \parallel x- \text {axis} — $L \parallel x- \text {axis}$

(ii) If a line is not parallel to $x$ -axis then let it intersect $x$ -axis at a point $P$ .

Inclination $\theta$ of $L$ is given by $\theta =m < APx$

(iii) Inclination of $\overline{AB},\overrightarrow{AB}$ or $\overrightarrow{BA}$ is defined as the inclination of $\overleftrightarrow{AB}$

Note:

$\to$ If $\theta$ is the inclination of a line then $0 \leq \theta < \pi$ .

$\to$ Parallel lines have same inclination and conversely.

$\to$ Inclination of perpendicular line differ by $\dfrac{\pi}{2}$ .

$\to$ Inclination is essentially an angle-measure. The only difference is that inclination can be zero, where as angle-measure can not be zero.

Slope (Gradient) of Non-Vertical Line:

The slope of a non-vertical line is given by $m=\tan \theta$ , where $\theta$ is the inclination of the line.

Note:

$\to$ Inclination $\theta \in [0,\pi]$ , but slope $\in R$ .

$\to$ Slope of a vertical line is not defined.

$\to$ Slope of a line is positive, zero or negative according as the inclination of the line is less than $\dfrac{\pi}{2}$ , equal to $\dfrac{\pi}{2}$ or greater than $\dfrac{\pi}{2}$ respectively.

$\to$ It is obvious that slope of a line is the ration of its rise or fall $(y_2-y_1)$ to its run $(x_2-x_1)$

A line with positive slope looks like rising where a line with negative slope looks like falling as we move from left to right along the line.

$\text { Slope} (m) =\dfrac{y_2-y_1}{x_2-x_1}$

$\to \dfrac{y_2-y_1}{x_2-x_1}=\dfrac{\triangle y}{\triangle x}$

Positive Slope:

Rise= Positive ; Run =Positive

Negative Slope:

Rise= Negative ; Run = Positive

Zero Slope:

Rise =0 ; Run =Positive

Undefined Slope:

Rise=Positive or Negative; Run =0