Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Lines and Planes in Space

Position of a line in relation to a plane


  1. A line  L   is said to lie in/on a plane  \lambda if  L \subset \lambda .
  2. A line  L  said to intersect a plane   \lambda if it does not lie on  \lambda   and has a point in common with the plane i.e.  \text {L} \not \subset \lambda \text { and } \text {L} \cap \lambda \neq \phi
 \overleftrightarrow{PQ} \text { on } \lambda
 L intersect  \lambda at 'P'
 \overline{PQ} \text { in } \lambda


It can be proved that \text {L}\cap \lambda contains exactly one point if \text {L}\cap \lambda \neq \phi and  \text {L} \not \subset \lambda  i.e. if  a line intersects a plane, it does so at exactly one point. For ,if  P \text { and } Q are distinct points in  \text {L} \cap \lambda, then clearly  \text { L}= \overleftrightarrow{PQ} \subset \lambda. Therefore  \text {L} lies on  \lambda which is impossible by the definition of intersection. The definition demands that the line must not lie on the plane.

3.  A line \text { L} , not lying on plane  \lambda is parallel to it, written as  \text {L} \parallel \lambda , if it has no point in common with the plane i.e. \text {L}\cap \lambda= \phi .

\text {L}\parallel \lambda


(i) It follows from the definitions of intersection and parallelism that a line must either intersect a plane or be parallel to it if it does not lie on the plane.

(ii) If a point  P or a line  L lies on a plane, it is also said that the planes passes through  P \text { or } L .

Lines in Space

(i) Intersecting Lines:

      Lines  L_1 \text { and } L_2 intersect each other if they are distinct (not coincident) and  \text {L}_1 \cap \text {L}_2 \neq \phi.


It proved that there is only one common two intersecting lines.

(ii) Parallel Lines

Distinct lines  \text {L}_1 \text { and } \text {L}_2 are called parallel if they are co-planer and have no point in common.

 L_1 \parallel L_2 \{L_1, L_2\} \subset \lambda

(iii) Skew Lines

A pair of non-coplanar lines are called skew lines.


(a) There is exactly one plane passing through two intersecting lines.

(b) There is exactly one plane passing through two parallel lines.

Skew lines occur only in a three-dimensional space.

Planes in Space

(I) Parallel Planes

      Distinct planes  \lambda_1 \text { and } \lambda_2 are parallel, written as  \lambda_1 \parallel \lambda_2 , if they have no point in common, i.e..  \lambda_1 \cap \lambda_2=\phi.

(ii) Intersecting Planes

Planes \lambda_1 \text { and }\lambda_2 are said to be intersecting if they are distinct and  \lambda_1 \cap \lambda_2 \neq \phi.

If two planes intersect, they must intersect along exactly one line, i.e. there can’t be more than one line common to two different planes.


There is exactly one plane containing two distinct parallel lines, for otherwise, there shall be two lines common to more than one number of planes.

Perpendicular (Normal) to a Plane:


A line intersecting a plane at a point  P is said to be perpendicular to the plane at   P  if it is perpendicular to every line lying on the plane which passes through  P .

 L \perp \lambda \text { i.e.,} L \perp \overleftrightarrow{AB}

Properties of  Line and Planes in Space

(I) Fact 1:

If a line  L is perpendicular to two intersecting lines  L_1 \text { and }  L_2  at their point of intersection, (say  P ) then  L is perpendicular to the plane of  L _1 \text { and } L_2 \quad at \quad P  .

'P' is called foot of perpendicular upon the plane.

(II) Fact 2:

There is exactly one line perpendicular to a plane at a given point on it. Also there is exactly one line perpendicular to a plane from an external point.

\to L is perpendicular to  \lambda at  P.  \to A is an external point \overline{AB} \perp \lambda ,  B lies in  P

(III) Fact 3:

 \overleftrightarrow{PQ} is perpendicular to a plane  \lambda at a point  Q on it.  L is a line on plane \lambda  and it does not pass through  Q  . If  \overleftrightarrow{QR} is perpendicular to  L \quad \text { at } \quad R, then  \overleftrightarrow{PR}  is also perpendicular to  L .

(IV) Fact 4:

The perpendicular in space to a line at a given point on it lies on exactly one plane which is perpendicular to the lines at the point. ( If a line is perpendicular to a plane, the plane is also called the perpendicular to the line.

(V) Fact 5:

Any two perpendiculars to a plane are a parallel.

\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}

(VI) Fact 6:

Two parallel lines, if one is perpendicular to a plane, then the other is also perpendicular to the same plane.

\left( \therefore L_1 \parallel L_2, \quad L_1 \perp \lambda \implies L_2 \perp \lambda \right)

Set of Parallel lines in Space:

A set of lines are called parallel if any two of them are parallel.

Common Transversal:

If a line  L intersects a set of lines  \{ L_1, L_2,...,L_n \}  then  L is called a common transversal of  L_1, L_2, ..., L_n .

A set of parallel or concurrent lines in space, having a common transversal are coplanar.

If  L_1, L_2, \quad \text { and } \quad L_1, L_2 are intersecting pairs of lines in space such that, L_1 \parallel L_1' \text { and } L_2 \parallel L_2' , then angle between  L_1, L_2 \text { and } L_1',L_2' are of equal measure.

 m < A0B=m < A' 0'B'=\theta
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