Introduction Of Three Dimensional Geometry Class 11 Notes Mathematics Chapter 12 - CBSE

Chapter : 12

What Are Introduction Of Three Dimensional Geometry ?

Coordinate Axes

In three-dimensions, the coordinate axes of a rectangular Cartesian coordinate system are three mutually perpendicular lines. The 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.

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Coordinate Of A Point In Space

The coordinates of a point P in the space are the perpendicular distances from P on three mutually perpendicular coordinates 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 the origin are O(0, 0, 0). The coordinate of any point on the X-axis will be as (x, 0, 0) and the coordinates of any point in the YZ plane will be as (0, y, z).

Distance Between Two Points

The distance between two points P(x1 , y1 , z1) and Q(x2 , y2 , z2) is given by

$$\text{PQ} =\\\sqrt{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2} + (z_{2} - z_{1})^{2}}$$

The distance of a point P(x, y, z) from the origin O(0, 0, 0) is given by

$$\text{OP =}\sqrt{x^{2} + y^{2} + z^{2}}$$

Section Formula

The coordinates of a point R which divides the lin e segment joining two points P(x1, y1 , z1) and Q(x2 , y2 , z2) internally and externally in the ratio m : n are given by

$$\bigg(\frac{mx_{2} +nx_{1}}{m+n},\frac{my_{2} +ny_{1}}{m+n},\\\frac{mz_{2} + nz_{1}}{m+n}\bigg)\\\text{and}\\\bigg(\frac{mx_{2}-nx_{1}}{m-n},\frac{my_{2}-ny_{1}}{m-n},\\\frac{mz_{2}-nz_{1}}{m-n}\bigg)\\\text{respectively}.$$

Coordinates Of Mid-point

The coordinates of mid-point of the line segment joining two points (x1 , y1 , z1) and (x2 , y2, z2 ) are

$$\bigg(\frac{x_{2} + x_{1}}{2},\frac{y_{2} + y_{1}}{2},\frac{Z_{2} + Z_{1}}{2}\bigg)$$

Centroid

The coordinates of centroid of the triangle whose vertices are (x1 , y1 , z1), (x2 , y2 , z2) and (x3, y3 , z3) are

$$\bigg(\frac{x_1 + x_{2} + x_{3}}{3},\frac{y_1 + y_{2} + y_{3}}{3},\\\frac{z_1 + z_{2} + z_{3}}{3}\bigg)$$