Maths > Three Dimensional Coordinate System > 2.0 Equation of a line in space
Three Dimensional Coordinate System
1.0 Introduction
2.0 Equation of a line in space
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
3.0 Distance and Angle between lines and 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
4.0 Plane
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
5.0 Relation between Plane, Line and Point.
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
6.0 Intersection of a line and a plane
7.0 Image of a point in a plane
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
3.2 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
4.2 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
5.2 Angle between a line and a plane
5.3 Distance of a point from a plane
5.4 Distance between two parallel planes
In order to find the equation of a line in vectorial form, we must know the position vector of a given point through which a line pass and a given vector parallel to the line.
Let us assume $\overrightarrow a $ be the position vector of a given point $A$ through which a line pass and $\overrightarrow b $ be the given vector parallel to the line.
Let $\overrightarrow r $ be the position vector of any arbitrary point $P$ on the line as shown in figure. Therefore, $\overrightarrow {AP} $ is parallel to $\overrightarrow b $ which can be written as $$\overrightarrow {AP} = \lambda b$$
where $\lambda $ is some real number.
But from figure we can say that, $$\overrightarrow {OP} = \overrightarrow {OA} + \overrightarrow {AP} = \overrightarrow a + \lambda \overrightarrow b $$ Hence, Vectorial form of a line is $$\overrightarrow r = \overrightarrow a + \lambda \overrightarrow b $$
