Three-Dimensional Geometry Fundamentals

Perpendicular Distance of a Point from Coordinate Planes

• Distance of a point $P(x, y, z)$ from the $xy\text{ plane}$ is equal to the absolute value of its $z$-coordinate:   $d_{xy} = |z|$

• Distance of a point $P(x, y, z)$ from the $yz\text{ plane}$ is equal to the absolute value of its $x$-coordinate:   $d_{yz} = |x|$

• Distance of a point $P(x, y, z)$ from the $zx\text{ plane}$ is equal to the absolute value of its $y$-coordinate:   $d_{zx} = |y|$

Distance Formula in Three Dimensions

• For any two points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ in three-dimensional space, the straight-line distance $d$ between $A$ and $B$ is given by:   $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Section Formula in Three Dimensions

• Consider two points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$. Let a point $C(x, y, z)$ divide the line segment joining $A$ and $B$ in the ratio $m : n$.

• Internal Division:   When the point $C(x, y, z)$ divides the line segment $AB$ internally in the ratio $m : n$, its coordinates are:   $x = \frac{mx_2 + nx_1}{m + n}$   $y = \frac{my_2 + ny_1}{m + n}$   $z = \frac{mz_2 + nz_1}{m + n}$

• External Division:   When the point $C(x, y, z)$ divides the line segment $AB$ externally in the ratio $m : n$, its coordinates are:   $x = \frac{mx_2 - nx_1}{m - n}$   $y = \frac{my_2 - ny_1}{m - n}$   $z = \frac{mz_2 - nz_1}{m - n}$