Introduction to Three-dimensional Geometry

3D Geometry

To locate position of a object in a line, you just need the distance of the object from the point of reference.

To locate the position of a point in a plane, we need two intersecting mutually perpendicular lines in the plane. These lines are called the coordinate axes and the two numbers are called the coordinates of the point with respect to the axes.

In actual life, we do not have to deal with points lying in a line or plane only.

For example, consider the position of a ball thrown in space at different points of time or the position of an aero plane as it flies from one place to another at different times during its flight.

If we were to locate the position of a ball thrown in the room (in the air) ,  we will not only re quire the perpendicular distances of the point to be located from two perpendicular walls of the room but also the height of the point from the floor of the room. Therefore, we need not only two but three numbers representing the perpendicular distances of the point from three mutually perpendicular planes, namely the floor of the room and two adjacent walls of the room. The three numbers representing the three distances are called the coordinates of the point with reference to the three coordinate planes. So, a point in space has three coordinates.

Coordinate Axes
In three dimensions, the coordinate axes of a rectangular cartesian coordinate system are three mutually perpendicular lines. These axes are called the X, Y and Z axes.

Coordinate Planes
The three planes determined by the pair of axes are the coordinate planes. These planes are called XY, YZ and ZX plane and they divide the space into eight regions known as octants.

Coordinates of a Point in Space
The coordinates of a point in the space are the perpendicular distances from P on three mutually perpendicular coordinate planes YZ, ZX, and XY respectively. The coordinates of a point P are written in the form of triplet like (x, y, z).
The coordinates of any point on

·         X-axis is of the form (x, 0,0)

·         Y-axis is of the form (0, y, 0)

·         Z-axis is of the form (0, 0, z)

·         XY-plane are of the form (x, y, 0)

·         YZ-plane is of the form (0, y, z)

·         ZX-plane are of the form (x, 0, z)

Distance Formula
The distance between two points P(x1, y1, z1) and Q(x2, y2, z2) is given by
Introduction to Three Dimensional Geometry Class 11 Notes Maths Chapter 12 1

The distance of a point P(x, y, z) from the origin O(0, 0, 0) is given by
OP = \sqrt { { x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 } }

Section Formula
The coordinates of the point R which divides the line segment joining two points P(x1, y1, z1) and Q(x2, y2, z2) internally or externally in the ratio m : n are given by
Introduction to Three Dimensional Geometry Class 11 Notes Maths Chapter 12 2

The coordinates of the mid-point of the line segment joining two points P(x1, y1, z1) and Q(x2, y2, z2) are
Introduction to Three Dimensional Geometry Class 11 Notes Maths Chapter 12 3

The coordinates of the centroid of the triangle, whose vertices are (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) are
Introduction to Three Dimensional Geometry Class 11 Notes Maths Chapter 12 4