Chapter 5: Three Dimensional GeometryClass 12 Mathematics — summary, notes, extra questions & MCQ quiz

Summary

This chapter extends coordinate geometry to three dimensions, describing lines and planes vectorially and in Cartesian form. A line is given by \(\vec{r}=\vec{a}+\lambda\vec{b}\), where \(\vec{a}\) is a point on it and \(\vec{b}\) its direction; in Cartesian form this is \(\dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}\). The angle between two lines comes from the dot product of their direction vectors. A plane is written as \(\vec{r}\cdot\hat{n}=d\), where \(\hat{n}\) is the unit normal, or in Cartesian form \(ax+by+cz=d\). The chapter computes the angle between two planes, between a line and a plane, and the shortest distance between two skew lines. These results give a complete algebraic description of position and orientation in space.

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