# Three Dimensional Geometry For Class 12

Three Dimensional Geometry for class 12 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, the angle between line and plane, between two lines etc. You need to practice many Three Dimensional Geometry questions to understand the topic better.

#### Direction Cosine and Direction Ratios of a line

Consider a line L passing through origin makes an angle of $\alpha, \beta, \gamma$ with x, y, and z-axes respectively, then the 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:

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

### 3d Geometry Class12 Important Questions

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