- Books Name
- Mathmatics Book Based on NCERT
- Publication
- KRISHNA PUBLICATIONS
- Course
- CBSE Class 12
- Subject
- Mathmatics
Concepts of Lines and Cartesian equation and vector equation of a line:
It is known that we can uniquely determine a line if:
- It passes through a particular point in a specific direction, or
- 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
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
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 :