Three dimensional Geometry covers few of the most important topics such as direction cosine and direction ratios of a line joining two points and also discusses the equation of lines and planes in space under different condition, angle between line and plane, between two line etc.

**Direction cosine and direction ratios of a line-**

Consider a line L passing through origin makes and angle of \(\alpha, \beta ,\gamma\) with x, y, and z axes respectively, then cosine of these angles is the direction cosine of the directed line L.

Any three numbers which are proportional to the direction cosines of a line are called the direction ratios of the line. Consider the direction cosine of line L be l,m,n and direction ratio a,b,c then \(a = \lambda l\), \(b = \lambda m\), \(c = \lambda n\), for non-zero \(\lambda \in R\).

\(\frac{l}{a} = \frac{m}{b} = \frac{n}{c} = k\)

Then the direction cosine are:

\(l = \pm \frac{a}{a^{2} + b^{2} + c^{2}}\), \(m = \pm \frac{b}{a^{2} + b^{2} + c^{2}}\), \(n = \pm \frac{c}{a^{2} + b^{2} + c^{2}}\)