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

Definition:

  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

Note:

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

Note:

(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.

Note:

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.

Notes:

(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.

Note:

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:

Definition:

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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