Introduction to Three Dimensional Geometry

• Coordinate Systems in 3D Space

  • Cartesian Coordinate System
    • Origin ($O$) and Three Mutually Perpendicular Axes ($X$-axis, $Y$-axis, $Z$-axis)
    • Coordinate Planes ($XY$-plane, $YZ$-plane, $ZX$-plane) dividing space into Octants (8 regions)
  • Coordinates of a Point
    • Represented as an ordered triplet $P(x, y, z)$
    • Projections of $P$ on coordinate planes and axes

• Distance between Two Points

  • Distance Formula in 3D Space
    • $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$
  • Distance from Origin
    • Distance of $P(x, y, z)$ from origin $O(0, 0, 0)$ is $\sqrt{x^2 + y^2 + z^2}$

• Section Formula

  • Internal Division
    • Coordinates of point dividing join of $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in ratio $m:n$ internally
    • Formula: $\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right)$
  • External Division
    • Coordinates for division in ratio $m:n$ externally (replace $+$ with $-$)
  • Mid-Point Formula
    • Special case where ratio is $1:1$
    • Formula: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)$

• Centroid of a Triangle and Tetrahedron

  • Centroid of a Triangle
    • Given vertices $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$, and $(x_3, y_3, z_3)$
    • Formula: $\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3}\right)$
  • Centroid of a Tetrahedron
    • Average of $x, y, z$ coordinates of all 4 vertices divided by 4