ANGLE BETWEEN TWO LINES

Consider two lines:

image

y = {m_1}x + {C_1}

and y = {m_2}x + {C_2} , where {m_1} = \tan {\theta _1} , {m_2} = \tan {\theta _2}

Angle between both lines are \theta \,{\rm{and}}\,\left( {180^\circ \, - \,\theta } \right).

\begin{array}{l}
{\rm{In }}\Delta {\rm{PAB,}}\\
\,\,\,\,\,\,\theta  + {\theta _2} = {\theta _1}\\
 \Rightarrow \theta  = {\theta _1} - {\theta _2}
\end{array}

Taking tan both sides,

\,\,\tan \theta  = \tan \left( {{\theta _1} - {\theta _2}} \right)

 \Rightarrow \,\tan \theta  = \frac{{\tan {\theta _1} - \tan {\theta _2}}}{{1 + \tan {\theta _1}\tan {\theta _2}}} = \left( {\frac{{{m_1} - {m_2}}}{{1 + {m_1}{m_2}}}} \right)

 \Rightarrow \,\tan \left( {180^\circ  - \theta } \right) =  - \left( {\frac{{{m_1} - {m_2}}}{{1 + {m_1}{m_2}}}} \right) 

\therefore \,\,{\rm{Angle}}\,{\rm{between}}\,{\rm{two}}\,{\rm{lines}},

\theta  = {\tan ^{ - 1}}\left| {\frac{{{m_1} - {m_2}}}{{1 + {m_1}{m_2}}}} \right|

 

Example

Find the angle between the lines7x - y = 1{\rm{ and }}6x - y = 11.

 

Solution

The equation is written as follows:

y = 7x - 1{\rm{ and }}y = 6x - 11

The angle between two lines is:

\tan \theta  = \left| {\frac{{{m_1} - {m_2}}}{{1 + {m_1}{m_2}}}} \right|

we know that{m_1} = 7,{m_2} = 6, substitute the values as follows:

\begin{array}{c}
\tan \theta  = \left| {\frac{{7 - 6}}{{1 + \left( 7 \right)\left( 6 \right)}}} \right|\\
 = \left| {\frac{1}{{1 + 42}}} \right|\\
 = \left| {\frac{1}{{43}}} \right|
\end{array}

Therefore, the angle between two lines is\theta  = \,{\tan ^{ - 1}}\frac{1}{{43}}.

CONDITIONS FOR PARALLELISM AND PERPENDICULARITY OF TWO LINES

Parallel lines definition: Parallel lines are distinct lines lying in the same plane and they never intersect each other. They have same slope.

Consider the figure as shown below:

clip_image032

Here the line RS and PQ are parallel.

Perpendicular lines definition:  Perpendicular lines are lines that intersect at right angles. Suppose if two lines are perpendicular to each other, then the product of their slope is always equal to –1.

Consider the figure as shown below:

image

Here the line AB and EF are perpendicular to each other.

Conditions of parallel and perpendicular lines

1. When the line is horizontal, the slope of the line is always zero.

2. When the line is vertical, the slope of the line is undefined.

3. When two lines are parallel, the slopes are equal.

4. When the two lines are perpendicular, the product of the slope is -1.

Consider two lines:

\begin{array}{c}
y = {m_1}x + {c_1}.....\left( 1 \right)\\
y = {m_2}x + {c_2}.....\left( 2 \right)
\end{array}

If line (1) is parallel to line (2), then, the slope{{m_1} = {m_2}}

If line (1) is perpendicular to line (2), then{m_1}{m_2} =  - 1.

Post Author: E-Maths