Question

Find the acute angle between the pair of lines \[x + 3y + 5 = 0\] and \[2x + y - 1 = 0\] .

Answer 1

Hint: Simplify the above given equations in \[y = mx + c\] format and find slopes of both lines. After that use the formula \[\tan \theta = \left| {\dfrac{{{m_1} - {m_2}}}{{1 + {m_1}{m_2}}}} \right|\] to get the angles between the lines where $m_1$ and $m_2$ are the slopes of lines.

Complete step-by-step answer:
The equation of first line is
 \[x + 3y + 5 = 0\]
 \[
  \therefore 3y = - x - 5 \\
  \therefore y = - \dfrac{x}{3} - \dfrac{5}{3} \\
 \]
On comparing the above equation by \[y = mx + c\] , we get the slope of the first line as \[{m_1} = - \dfrac{1}{3}\] .
The equation of second line is
 \[2x + y - 1 = 0\]
 \[\therefore y = - 2x + 1\]
On comparing the above equation by \[y = mx + c\] , we get the slope of the second line as \[{m_2} = - 2\] .
Now, for angle between the lines, we use the formula \[\tan \theta = \left| {\dfrac{{{m_1} - {m_2}}}{{1 + {m_1}{m_2}}}} \right|\] .
 \[\therefore \tan \theta = \left| {\dfrac{{ - \dfrac{1}{3} - \left( { - 2} \right)}}{{1 + \left( { - \dfrac{1}{3}} \right)\left( { - 2} \right)}}} \right|\]
 \[\therefore \tan \theta = \left| {\dfrac{{ - \dfrac{1}{3} + 2}}{{1 + \dfrac{2}{3}}}} \right|\]
 \[\therefore \tan \theta = \left| {\dfrac{{\dfrac{{ - 1 + 6}}{3}}}{{\dfrac{{3 + 2}}{3}}}} \right|\]
 \[
  \therefore \tan \theta = \left| {\dfrac{5}{5}} \right| \\
  \therefore \tan \theta = \left| 1 \right| \\
  \therefore \theta = {\tan ^{ - 1}}\left( 1 \right) \\
  \therefore \theta = 45^\circ \\
 \]
Thus, the angle between the lines is \[45^\circ \] .

Note: The slope of a line can also be written as \[m = \tan \theta \] .
So, here \[{m_1} = \tan {\theta _1} = - \dfrac{1}{3}\] and \[{m_2} = \tan {\theta _2} = - 2\] .
Instead of \[\tan \theta = \left| {\dfrac{{{m_1} - {m_2}}}{{1 + {m_1}{m_2}}}} \right|\] , you can also use the formula \[\tan \left( {{\theta _1} - {\theta _2}} \right) = \left| {\dfrac{{\tan {\theta _1} - \tan {\theta _2}}}{{1 + \tan {\theta _1}\tan {\theta _2}}}} \right|\] to find the angle between the lines.