Chapter 11: Introduction to Three-Dimensional Geometry

In Class XI you have so far worked in the plane: two coordinates $(x, y)$ name a point. The real world is three-dimensional, and we now add a third coordinate, $z$. A point in space is written $(x, y, z)$. Three perpendicular coordinate axes , the $x$, $y$, and $z$-axes , split space into eight octants (the 3D analogue of the four quadrants).

The good news: most of what you learned in 2D extends with one extra coordinate.

• Distance between $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$: $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$.
• Section formula internal in $m : n$: $\left(\dfrac{m x_2 + n x_1}{m+n}, \dfrac{m y_2 + n y_1}{m+n}, \dfrac{m z_2 + n z_1}{m+n}\right)$.
• Midpoint: average each coordinate.
• Collinearity: equal ratios of differences in each coordinate.

This chapter is short , only the foundations of 3D coordinate geometry. In Class XII you will build on it with vectors, direction cosines, lines, planes, and the angle between them. For now, master the basics of locating points in space and measuring between them.

What's inside

1. Coordinates in space and octants , naming and locating points.
2. Distance between two points in 3D.
3. Section formula in 3D.
4. Midpoint, centroid, and equidistant problems.
5. Collinearity of three points and equation of a line preview.
6. Miscellaneous: locus and applications.

Key results / Formula card

How to read this chapter

Visualise. Hold up your right hand: thumb is $x$, index $y$, middle finger $z$. Every problem can be sketched roughly. The algebra is identical to the 2D case, with one extra term , but spatial intuition is the real prize.