Concepts of Lines and Cartesian equation and vector equation of a line:

It is known that we can uniquely determine a line if:

1. It passes through a particular point in a specific direction, or
2. It passes through two unique points

Equation of a Line passing through a point and parallel to a vector:

Let us consider that the position vector of the given point be     with respect to the origin.

The line passing through point A is given by l and it is parallel to the vector

as shown below. Let us choose any random point R on the line l and its position vector with respect to origin of the rectangular co-ordinate system is given by

It is vector equation of line passing through a point and parallel to a vector.

Cartesian form:

If the three-dimensional co-ordinates of the point ‘A’ are given as (x₁, y₁, z₁) and the direction cosines of this point is given as a, b, c then considering the rectangular co-ordinates of point P as (x, y, z):

Let

It is cartesian equation of line passing through a point and parallel to a vector.

Equation of a Line passing through two unique given point:

Let us consider that the position vector of the given two point be      and  with respect to the origin.

Let us choose any random point P on the line and its position vector with respect to origin of the rectangular co-ordinate system is given by  .

Point P lies on the line AB if and only if the vectors    

It is vector equation of line passing through two point .

Cartesian Form:

It is the Cartesian equation of a line. passing through two given points.

Example:

Find the vector and Cartesian equations of the line passing through the points A(3,4,−6) and B(5,−2,7)

Solution:

Example:

Write vector and the cartesian equations of the lines that passes through the origin and (5,−2,3)

Solution:

 

The line passing through (0,0,0) and (5,−2,3)

Example:

Find the cartesian equation of the line which passes through the points (7,4,6) and (9,1,8).

Solution:

Here , (x₁,y₁,z₁) = (7,4,6) and (x₂,y₂,z₂) = (9,1,8)

Direction ratios of the line are x₂​−x₁ = 2, ​y₂​−y₁ = -3 , ​z₂​−z₁​​=2​

Equation of line :