A line $L$ is said to lie in/on a plane $\lambda$ if $L \subset \lambda$ .
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. $\text {L} \not \subset \lambda \text { and } \text {L} \cap \lambda \neq \phi$

$\overleftrightarrow{PQ} \text { on } \lambda$

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. $\text {L}\cap \lambda= \phi$ .

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.

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