Three dimensional geometry

Three Dimensional Geometry

Direction Cosines and Direction Ratios of

a Line

Direction ratio also known as Direction Numbers

Direction Cosine of a line are

Direction Cosine of a Line

Direction Cosine of a Line

Passing through two points

Direction Ratios of the line segment

Example

Example

A Line is uniquely determined if

It passes through a given point and

has given direction.

It passes through two given points

Equation of a Line in Space

Equation of a line through a given point and parallel to a given vector. Vector Equation of the line is given by

Equation of a Line passing through a point Vector

form

Cartesian equation of

the line is

Equation of a Line passing through a point Cartesian

form

Equation of a Line passing through two given points

Angle between two Lines

Example

If two lines in space intersect at a point, then

the shortest distance between them is zero.

If two lines in space are parallel, then the shortest distance between them will be the perpendicular distance.

In a space, there are lines which are neither intersecting nor parallel. Such a pair of lines are non coplanar and are called skew lines.

Skew Lines

The shortest distance between the skew lines l1

and With equations

Shortest Distance between Skew Lines

Distance between the

Parallel Lines

The vector form of the equation of the plane is

In Cartesian Form

Equation of a Plane in a Normal Form

Note

Cartesian form

Vector form

Equation of a plane perpendicular to a given

vector and passing through a point

Vector Form

Cartesian Form

Equation of a plane passing through three non

collinearpoints

Intercept form of the equation of a plane

Vector Form

Cartesian Form

Plane Passing through intersection of two planes

Co planarity of two

Lines

Distance of a point

from a Plane

Vector form

Cartesian form

Distance of a point from a Plane

Angle between a line and a

plane