# Three Dimensional Geometry For Class 12

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:

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

#### Practise This Question

A random variable X has the following probability distribution
XP(X=x)XP(X=x)0λ511λ13λ613λ25λ715λ37λ817λ49λ
then, λ is equal to