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Introduction To 3-D Geometry class 11 Notes Mathematics

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CBSE Class 11 Mathematics
Revision Notes
Chapter-12
Introduction To 3-D Geometry class 11 Notes Mathematics

Coordinates- axes, planes, points in 3D
Distance between Two Points
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.

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

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

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

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 ${\text{Any point on}}\;{\text{XY }} \to {\text{ plane}}\;\;\left( {{\text{ x}},{\text{ y}},{\text{ }}0} \right)$ ${\text{Any point on}}\;{\text{YZ }} \to {\text{plane}}\;\;\;\;\left( {{\text{ }}0,{\text{ y}},{\text{ z}}} \right)$ ${\text{Any point on}}\;{\text{ZX }} \to {\text{plane}}\;\;\;\left( {{\text{ x}},{\text{ }}0,{\text{z}}} \right)$

Distance formula between two points: Distance between two points $\text{P }\left( \text{ }{{\text{x}}_{1}}\text{ },\text{ }{{\text{y}}_{1}},\text{ }{{\text{z}}_{1}}\text{ } \right)\text{ and}~\text{Q }\left( \text{ }{{\text{x}}_{2}}\text{ },\text{ }{{\text{y}}_{2}}\text{ },{{\text{z}}_{2}}\text{ } \right)\text{ is}~$

$\left| \text{PQ} \right|$ $\text{ }=\text{ }\sqrt{{{\left( ~{{x}_{2}}-{{x}_{1}}~ \right)}^{2}}+~\left( ~{{y}_{2}}-{{y}_{1}}{{)}^{2}} \right)+~{{\left( ~{{z}_{2}}-{{z}_{1}} \right)}^{2}}}~$

Section Formula: The co-ordinates of R which divides a line segment joining the points $~~~~\text{P }\left( \text{ }{{\text{x}}_{1}}\text{ },\text{ }{{\text{y}}_{1}},\text{ }{{\text{z}}_{1}}\text{ } \right)\text{ and}~\text{Q }\left( \text{ }{{\text{x}}_{2}}\text{ },\text{ }{{\text{y}}_{2}},\text{ }{{\text{z}}_{2}}\text{ } \right)~$

Internally and externally in the ratio m : n respectively

Internally: $R\left( \frac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\frac{m{{y}_{2}}+n{{y}_{1}}}{m+n},\frac{m{{z}_{2}}+n{{z}_{1}}}{m+n} \right)$

Externally: $S\left( \frac{m{{x}_{2}}-n{{x}_{1}}}{m-n},\frac{m{{y}_{2}}-n{{y}_{1}}}{m-n},\frac{m{{z}_{2}}-n{{z}_{1}}}{m-n} \right)$

Centroid: The coordinates of the centroid of the trinagle whose vertices are $~~~~\left( \text{ }{{\text{x}}_{1}}\text{ },\text{ }{{\text{y}}_{1}},\text{ }{{\text{z}}_{1}}\text{ } \right)\text{ }$

$~\left( \text{ }{{\text{x}}_{2}}\text{ },\text{ }{{\text{y}}_{2}},\text{ }{{\text{z}}_{2}}\text{ } \right)\ \ \ ~and\ \ \ ({{x}_{3}},\ {{y}_{3}},{{z}_{3}})$ is

$\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3},\frac{{{z}_{1}}+{{z}_{2}}+{{z}_{3}}}{3} \right)$

Introduction To 3-D Geometry class 11 Notes

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