2.4 Cartesian form of a line passing through two given points

2.1 Vectorial form of a line passing through a given point and parallel to a given vector
2.2 Vectorial form of a line passing through two given points
2.3 Cartesian form of a line passing through a given point and parallel to a given vector
2.4 Cartesian form of a line passing through two given points

3.1 Condition for perpendicularity
3.2 Condition for parallelism
3.3 Co-planarity of two lines
3.4 Shortest distance between two lines
3.5 Distance between two skew lines
3.6 Distance between parallel lines
3.7 Perpendicular distance of a point from a line
3.8 Image of a point in a straight line

4.1 Equation of plane in normal form
4.2 Equation of a plane perpendicular to a given vector and passing through a given point
4.3 Equation of plane passing through a given point and parallel to the two given vectors
4.4 Equation of plane passing through three non-collinear points
4.5 Intercept form of plane
4.6 Equation of plane passing through the intersection of two given planes

5.1 Angle between two planes
5.2 Angle between a line and a plane
5.3 Distance of a point from a plane
5.4 Distance between two parallel planes

Let us consider the co-ordinates of any arbitrary point on line $P$ be $(x,y,z)$, the co-ordinates of given point $A$ and $B$ through which line passes be $({x_1},{y_1},{z_1})$ and $({x_2},{y_2},{z_2})$ respectively.

Therefore,

$$\vec r = x\hat i + y\hat j + z\hat k{\text{ ; }}\vec a = {x_1}\hat i + {y_1}\hat j + {z_1}\hat k{\text{ ; }}\vec b = {x_2}\hat i + {y_2}\hat j + {z_2}\hat k$$

Now, substitute these values in the vectorial form of line i.e.,

$$\overrightarrow r = \overrightarrow a + \lambda \left( {\overrightarrow b - \overrightarrow a } \right)$$ We get,

$$\begin{equation} \begin{aligned} \left( {x\hat i + y\hat j + z\hat k} \right) = \left( {{x_1}\hat i + {y_1}\hat j + {z_1}\hat k} \right){\text{ + }}\lambda \left\{ {\left( {{x_2}\hat i + {y_2}\hat j + {z_2}\hat k} \right) - \left( {{x_1}\hat i + {y_1}\hat j + {z_1}\hat k} \right){\text{ }}} \right\} \\ \left( {x\hat i + y\hat j + z\hat k} \right) = \left\{ {{x_1} + \lambda \left( {{x_2} - {x_1}} \right)} \right\}\hat i + \left\{ {{y_1} + \lambda \left( {{y_2} - {y_1}} \right)} \right\}\hat j + \left\{ {{z_1} + \lambda \left( {{z_2} - {z_1}} \right)} \right\}\hat k \\\end{aligned} \end{equation} $$ On comparing the co-efficients, we get

$$\begin{equation} \begin{aligned} x = {x_1} + \lambda \left( {{x_2} - {x_1}} \right) \\ y = {y_1} + \lambda \left( {{y_2} - {y_1}} \right) \\ z = {z_1} + \lambda \left( {{z_2} - {z_1}} \right) \\\end{aligned} \end{equation} $$

which are the parametric equations of the line.

By eliminating the parameter $\lambda $, we get,

$$\frac{{x - {x_1}}}{{{x_2} - {x_1}}} = \frac{{y - {y_1}}}{{{y_2} - {y_1}}} = \frac{{z - {z_1}}}{{{z_2} - {z_1}}}$$

which is the cartesian form of the line.