# Three-Dimensional Geometry

In three dimensions, the coordinate axes of a rectangular Cartesian co-ordinate system are three mutually perpendicular lines. The axes are called the x, y and z axes.

Co-ordinates of a Point: 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 planes.

• Any point on x-axis is of the form (x, 0, 0)
• Any point on y-axis is of the form (0, y, 0)
• Any point on z-axis is of the form (0, 0, z)

Distance between Two Points: Distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) is given by $$d=\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}+{{({{z}_{2}}-{{z}_{1}})}^{2}}}$$.Direction Cosines: Direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axes. If l, m, n are the direction cosines of a line, then l² + m² + n² = 1.Direction cosines of a line joining two points A (x₁, y₁, z₁) and B (x₂, y₂, z₂) are $$\frac{{{x}_{2}}-{{x}_{1}}}{AB}$$, $$\frac{{{y}_{2}}-{{y}_{1}}}{AB}$$, $$\frac{{{z}_{2}}-{{z}_{1}}}{AB}$$.

$$AB=d=\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}+{{({{z}_{2}}-{{z}_{1}})}^{2}}}$$.

Direction Ratios: Direction ratios of a line are the numbers which are proportional to the direction cosines of a line. If l, m, n are the direction cosines and a, b, c are the direction ratios of a line, then $$l=\frac{a}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}$$, $$m=\frac{b}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}$$, $$n=\frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}$$.

Skew Lines: Skew lines are lines in space which are neither parallel nor intersecting. They lie in different planes.

Angle between Skew Lines: Angle between skew lines is the angle between two intersecting lines drawn from any point (or origin) parallel to each of the skew lines.

cosθ = |l₁l₂ + m₁m₂ + n₁n₂|

$$\cos \theta \,=\,\left| \frac{{{a}_{1}}{{a}_{2}}\,+\,{{b}_{1}}{{b}_{2}}\,+\,{{c}_{1}}{{c}_{2}}}{\sqrt{a_{1}^{2}\,+\,b_{1}^{2}\,+\,c_{1}^{2}}\sqrt{a_{2}^{2}\,+\,b_{2}^{2}\,+\,c_{2}^{2}}} \right|$$