Maths > Three Dimensional Coordinate System > 5.0 Relation between Plane, Line and Point.
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
5.2 Angle between a line and a plane
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
The angle $\phi $ between a line and a plane can be considered as the complement of the angle $\theta $ between the parallel vector to the line and the normal vector to the plane.
Let us assume the vector equation of plane be $$\overrightarrow r .\overrightarrow n = d$$ where $\overrightarrow n $ is the normal vector to the plane and the vector equation of line be $$\overrightarrow r = \overrightarrow a + \lambda \overrightarrow b $$ where $\overrightarrow b $ is the parallel vector to the line. Therefore, the angle $\theta $ between the parallel vector to the line and the normal vector to the plane is $$\cos \theta = \left| {\frac{{\overrightarrow b .\overrightarrow n }}{{\left| {\overrightarrow b } \right|\left| {\overrightarrow n } \right|}}} \right|$$
Therefore, the angle $\phi $ between a line and a plane can be calculated by putting $$\phi = {90^ \circ } - \theta $$
$$\begin{equation} \begin{aligned} \cos \theta = \sin ({90^ \circ } - \theta ) \\ sin\phi = \left| {\frac{{\overrightarrow b .\overrightarrow n }}{{\left| {\overrightarrow b } \right|\left| {\overrightarrow n } \right|}}} \right| \\ \phi = {\sin ^{ - 1}}\left| {\frac{{\overrightarrow b .\overrightarrow n }}{{\left| {\overrightarrow b } \right|\left| {\overrightarrow n } \right|}}} \right| \\\end{aligned} \end{equation} $$
