### Mathematics Class XI

Unit-I: Sets and Functions
Chapter 1: Sets
Unit-II: Algebra
Chapter 5: Binomial Theorem
Chapter 6: Sequence and Series
Unit-III: Coordinate Geometry
Chapter 1: Straight Lines
Chapter 2: Conic Sections
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability
Chapter 1: Statistics
Chapter 2: Probability

# Lines and Planes in Space

Position of a line in relation to a plane

Definition:

1. A line is said to lie in/on a plane if .
2. A line said to intersect a plane if it does not lie on and has a point in common with the plane i.e. Note:

It can be proved that contains exactly one point if and i.e. if  a line intersects a plane, it does so at exactly one point. For ,if are distinct points in , then clearly . Therefore lies on which is impossible by the definition of intersection. The definition demands that the line must not lie on the plane.

3.  A line , not lying on plane is parallel to it, written as , if it has no point in common with the plane i.e. .

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 or a line lies on a plane, it is also said that the planes passes through .

Lines in Space

(i) Intersecting Lines:

Lines intersect each other if they are distinct (not coincident) and .

Note:

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

(ii) Parallel Lines

Distinct lines are called parallel if they are co-planer and have no point in common.

(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 are parallel, written as , if they have no point in common, i.e.. .

(ii) Intersecting Planes

Planes are said to be intersecting if they are distinct and .

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 is said to be perpendicular to the plane at if it is perpendicular to every line lying on the plane which passes through .

Properties of  Line and Planes in Space

(I) Fact 1:

If a line is perpendicular to two intersecting lines at their point of intersection, (say ) then is perpendicular to the plane of .

(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.

(III) Fact 3: is perpendicular to a plane at a point on it. is a line on plane and it does not pass through . If is perpendicular to then is also perpendicular to .

(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.

(VI) Fact 6:

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

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 intersects a set of lines then is called a common transversal of .

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

If are intersecting pairs of lines in space such that, , then angle between are of equal measure.