Position of a line in relation to a plane
Definition:
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.