Chapter 11: Three-Dimensional Geometry

In Class XI you met three-dimensional coordinates and basic distance formulas. This chapter takes the next step: the algebra of lines and planes in 3D, and the geometry that follows from it. Vector algebra from Chapter 10 becomes the natural language. Many results that look complicated in pure Cartesian form become one-line vector identities.

A line in 3D is determined by a point and a direction. Given a point $A$ with position vector $\vec a$ and a direction vector $\vec d$, the line consists of all points $\vec r = \vec a + t \vec d$ as $t$ ranges over $\mathbb{R}$. The same line can be written in Cartesian form as $\dfrac{x - x_0}{d_1} = \dfrac{y - y_0}{d_2} = \dfrac{z - z_0}{d_3}$. Both forms have their uses.

A plane is determined by a point and a normal direction (the perpendicular to the plane). With normal vector $\vec n$ and a point $\vec a$ on the plane, the plane consists of all $\vec r$ with $(\vec r - \vec a)\cdot \vec n = 0$, or equivalently $\vec r\cdot\vec n = d$ (where $d = \vec a\cdot\vec n$ is a constant). The Cartesian form is $Ax + By + Cz = D$, with $(A, B, C)$ the normal direction.

This chapter contains four families of problems: writing equations of lines and planes; computing angles between them; computing distances (point to line, point to plane, line to line); and finding intersections (line meets plane, two planes meet in a line, three planes meet in a point or not at all). The toolkit , dot products, cross products, triple products , is exactly Chapter 10's.

For board examinations you must be fluent in moving between vector and Cartesian forms. JEE problems often hinge on a clever vector setup , choosing a good normal direction, or recognising that two lines are skew rather than intersecting.

A note on conventions. Direction cosines are denoted $(l, m, n)$ with $l^2 + m^2 + n^2 = 1$. Direction ratios are any triple proportional to them. The angle between two lines is always taken in $[0, \pi/2]$ (use absolute value of cosine).

What's inside

1. Direction cosines, direction ratios, and lines , describing a line by a point and a direction.
2. Vector and Cartesian equations of a line , two forms of the same object.
3. Angle between two lines, perpendicularity, parallelism , vector formulas.
4. Shortest distance between two skew lines , the cross-product formula.
5. Vector and Cartesian equations of a plane , three standard forms.
6. Angle between a line and a plane, line in a plane , when does a line lie in a plane?
7. Distance from a point to a plane; intersections , point-to-plane formula and line-plane intersection.

Key results / Formula card

How to read this chapter

Be comfortable with both vector and Cartesian forms , board exams ask for one, JEE often the other. Sketch when possible. Remember the four "shapes": a line, a plane, intersections of two of these, and distances. Every problem reduces to a small computation in one of these categories.