Maths > Three Dimensional Coordinate System > 3.0 Distance and Angle between lines and points.
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
3.2 Condition for parallelism
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
If the two lines are parallel means $$\theta = {0^ \circ }$$ First of all, we have to find the angle between two lines in terms of $\sin \theta $ using the formulae $$\begin{equation} \begin{aligned} {\sin ^2}\theta + {\cos ^2}\theta = 1 \\ \Rightarrow \sin \theta = \sqrt {1 - {{\cos }^2}\theta } \\ {\text{ = }}\sqrt {1 - \frac{{{{\left( {{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}} \right)}^2}}}{{\left( {{a_1}^2 + {b_1}^2 + {c_1}^2} \right)\left( {{a_2}^2 + {b_2}^2 + {c_2}^2} \right)}}} \\ {\text{ = }}\frac{{\sqrt {{{\left( {{a_1}{b_2} - {a_2}{b_1}} \right)}^2} + {{\left( {{b_1}{c_2} - {b_2}{c_1}} \right)}^2} + {{\left( {{c_1}{a_2} - {c_2}{a_1}} \right)}^2}} }}{{\sqrt {{a_1}^2 + {b_1}^2 + {c_1}^2} \sqrt {{a_2}^2 + {b_2}^2 + {c_2}^2} }} \\\end{aligned} \end{equation} $$
Now, if two lines are parallel, then $$\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} = \frac{{{c_1}}}{{{c_2}}}$$
Note: Two parallel lines have same direction cosines i.e., $$\begin{equation} \begin{aligned} {l_1} = {l_2} \\ {m_1} = {m_2} \\ {n_1} = {n_2} \\\end{aligned} \end{equation} $$
