\displaystyle \textbf{1. Three-Dimensional Cartesian Coordinate System}
\displaystyle \text{In three dimensions, the coordinate axes of a rectangular Cartesian coordinate system are}
\displaystyle \text{three mutually perpendicular lines called the }x\text{-}axis,\;y\text{-}axis\text{ and }z\text{-}axis.
\\

\displaystyle \textbf{2. Coordinate Planes}
\displaystyle \text{The three planes determined by the pairs of axes are called the coordinate planes.}
\displaystyle \text{These planes are }xy,\;yz\text{ and }zx\text{ planes and they divide the space into eight regions}
\displaystyle \text{known as }octants.
\\

\displaystyle \textbf{3. Coordinates of a Point in Space}
\displaystyle \text{The coordinates of a point }P\text{ in space are the perpendicular distances from }P\text{ on the}
\displaystyle \text{three mutually perpendicular coordinate planes }YZ,\;ZX\text{ and }XY\text{ respectively.}
\displaystyle \text{The coordinates of }P\text{ are written as the triplet }(x,y,z).
\\

\displaystyle \textbf{4. Interpretation of Coordinates}
\displaystyle \text{The coordinates of a point are also the distances from the origin of the feet of the perpendiculars}
\displaystyle \text{from the point on the respective coordinate axes.}
\\

\displaystyle \textbf{5. Coordinates of Points on Axes and Planes}
\displaystyle \text{(i) Points on the }x\text{-}axis\text{ are of the form }(x,0,0)
\displaystyle \text{(ii) Points on the }y\text{-}axis\text{ are of the form }(0,y,0)
\displaystyle \text{(iii) Points on the }z\text{-}axis\text{ are of the form }(0,0,z)
\displaystyle \text{(iv) Points on the }xy\text{-}plane\text{ are of the form }(x,y,0)
\displaystyle \text{(v) Points on the }yz\text{-}plane\text{ are of the form }(0,y,z)
\displaystyle \text{(vi) Points on the }zx\text{-}plane\text{ are of the form }(x,0,z)
\\

\displaystyle \textbf{6. Distance Formula in Three Dimensions}
\displaystyle \text{The distance between two points }P(x_1,y_1,z_1)\text{ and }Q(x_2,y_2,z_2)\text{ is given by}
\displaystyle PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}
\\

\displaystyle \textbf{7. Distance from the Origin}
\displaystyle \text{The distance of a point }P(x,y,z)\text{ from the origin }O(0,0,0)\text{ is given by}
\displaystyle OP=\sqrt{x^2+y^2+z^2}
\\

\displaystyle \textbf{8. Section Formula}
\displaystyle \text{The coordinates of the point }R\text{ dividing the line segment joining }
\displaystyle P(x_1,y_1,z_1)\text{ and }Q(x_2,y_2,z_2)\text{ in the ratio }m:n\text{ are}
\displaystyle \left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n},\frac{mz_2+nz_1}{m+n}\right)
\displaystyle \text{internally, and}
\displaystyle \left(\frac{mx_2-nx_1}{m-n},\frac{my_2-ny_1}{m-n},\frac{mz_2-nz_1}{m-n}\right)
\displaystyle \text{externally.}
\\

\displaystyle \textbf{9. Mid-point Formula}
\displaystyle \text{The coordinates of the mid-point of the line segment joining }
\displaystyle (x_1,y_1,z_1)\text{ and }(x_2,y_2,z_2)\text{ are}
\displaystyle \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)
\\

\displaystyle \textbf{10. Centroid of a Triangle}
\displaystyle \text{The coordinates of the centroid of the triangle whose vertices are }
\displaystyle (x_1,y_1,z_1),\;(x_2,y_2,z_2)\text{ and }(x_3,y_3,z_3)\text{ are}
\displaystyle \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)
\\

Discover more from ICSE / ISC / CBSE Mathematics Portal for K12 Students

Subscribe to get the latest posts sent to your email.