Position of a line in relation to a plane
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. .
(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 .
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.
(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.
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 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.
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.