Introduction to 3D Geometry - Revision Notes

 CBSE Class 11 Mathematics

Revision Notes
Chapter-12
INTRODUCTION TO THREE DIMENSIONAL GEOMETRY

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1. Coordinates- axes, planes, points in 3D
2. Distance between Two Points
3. Section Formula

• 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-axis, y-axis and z-axis.
• Planes: The three planes determined by the pair of axes are the coordinate planes, called XY, YZ and ZX planes.

          xy−$xy-$plane i.e., z=0$z=0$

          yz−$yz-$plane i.e., x=0$x=0$

          zx−$zx-$plane i.e., y=0$y=0$

• Octants: The three coordinate planes divide the space into eight parts known as octants.
• Points in 3D: The coordinates of a point P in three dimensional geometry is always written in the form of triplet like (x, y, z). Here x, y  and z are the distances from the YZ, ZX and  XY

  Any point onXY → plane( x, y, 0)$\text{Any point on}\text{XY }\to \text{ plane}\left(\text{ x},\text{ y},0\right)$

  Any point onYZ →plane( 0, y, z)$\text{Any point on}\text{YZ }\to \text{plane}\left(0,\text{ y},\text{ z}\right)$

  Any point onZX →plane( x, 0,z)$\text{Any point on}\text{ZX }\to \text{plane}\left(\text{ x},0,\text{z}\right)$

• Distance formula between two points: Distance between two points P ( x1 , y1, z1 ) and Q ( x2 , y2 ,z2 ) is $\text{P }\left({\text{x}}_1,{\text{y}}_1,{\text{z}}_1\right)\text{ and}\text{Q }\left({\text{x}}_2,{\text{y}}_2,{\text{z}}_2\right)\text{ is}$

|PQ|$\left|\text{PQ}\right|$ = ( x2−x1 )2+ ( y2−y1)2)+ ( z2−z1)2−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√ $=\sqrt{{\left(x_2-x_1\right)}^2+\left(y_2-y_1{)}^2\right)+{\left(z_2-z_1\right)}^2}$

Section Formula: The co-ordinates of R which divides a line segment joining the points     P ( x1 , y1, z1 ) and Q ( x2 , y2, z2 ) $\text{P }\left({\text{x}}_1,{\text{y}}_1,{\text{z}}_1\right)\text{ and}\text{Q }\left({\text{x}}_2,{\text{y}}_2,{\text{z}}_2\right)$

Internally and externally in the ratio m : n respectively

Internally:          R(mx2+nx1m+n,my2+ny1m+n,mz2+nz1m+n)$R\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n},\frac{m{z}_2+n{z}_1}{m+n}\right)$

Externally:         S(mx2−nx1m−n,my2−ny1m−n,mz2−nz1m−n)$S\left(\frac{m{x}_2-n{x}_1}{m-n},\frac{m{y}_2-n{y}_1}{m-n},\frac{m{z}_2-n{z}_1}{m-n}\right)$

Centroid: The coordinates of the centroid of the trinagle whose vertices are     ( x1 , y1, z1 ) $\left({\text{x}}_1,{\text{y}}_1,{\text{z}}_1\right)$

 ( x2 , y2, z2 )    and   (x3, y3,z3)$\left({\text{x}}_2,{\text{y}}_2,{\text{z}}_2\right)and(x_3,y_3,z_3)$ is

(x1+x2+x33,y1+y2+y33,z1+z2+z33)