Equation of Line Perpendicular to another Line

Consider we have given equation of a line and this given line is perpendicular to another line which is passes through given point and to find the required equation of line with the help of given line.


Now we follow the following procedure with the help of an example

Example: Find the equation of straight line passing through the point \left( {1,2} \right) and is perpendicular to another given line whose equation is 2x -  3y + 5 = 0.

First find the slope of given line by comparing slope intercept form of a line as follows:

\begin{gathered} 2x - 3y + 5 = 0 \\ \Rightarrow 3y = 2x + 5 \\ \Rightarrow y = \frac{2}{3}x + \frac{5}{3} \\ \end{gathered}

Compare with slope intercept form, to find the slope of given line y = mx + c
Now slope of given line m = \frac{2}{3}, since given line is perpendicular to the required line under the condition of perpendicular lines {m_1} \times {m_2} =  - 1
Now slope of required line is m =  -  \frac{3}{2}, since the required line passes through the given point \left( {1,2} \right). To find equation of required line using slope point form is given by

\begin{gathered} y - {y_1} = m\left( {x - {x_1}} \right) \\ \Rightarrow y - 2 =  - \frac{3}{2}\left( {x - 1} \right) \\ \Rightarrow 2y - 4 =  - 3x + 3 \\ \Rightarrow 3x + 2y - 7 = 0 \\ \end{gathered}

Which is the equation of straight line perpendicular to the line 2x - 3y + 5 = 0. In this calculated equation we observe that the coefficients of x and y are cross with each other with negative sign between them.