Chapter 11 – Three-Dimensional Geometry (JEE)
1. Introduction to Three-Dimensional Geometry
Three-dimensional geometry deals with the study of points, lines and planes in
space using coordinates.
It extends two-dimensional coordinate geometry into space.
2. Coordinate System in Space
In 3D space, a point is represented by an ordered triplet $(x, y, z)$.
- X-axis → length
- Y-axis → breadth
- Z-axis → height
3. Distance Between Two Points
Distance between $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$:
$$PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$$
4. Section Formula (Internal Division)
Coordinates of point dividing line joining $A(x_1,y_1,z_1)$ and
$B(x_2,y_2,z_2)$ in ratio $m:n$:
$$\left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n},\frac{mz_2+nz_1}{m+n}\right)$$
5. Section Formula (External Division)
$$\left(\frac{mx_2-nx_1}{m-n},\frac{my_2-ny_1}{m-n},\frac{mz_2-nz_1}{m-n}\right)$$
6. Coordinates of Mid-Point
Mid-point of $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$:
$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)$$
7. Direction Cosines
Direction cosines $(l,m,n)$ are cosines of angles made by a line with X, Y, Z axes.
$l^2+m^2+n^2=1$
8. Direction Ratios
Any three numbers proportional to direction cosines are called direction ratios.
9. Angle Between Two Lines
If direction ratios are $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$:
$$\cos\theta=\frac{a_1a_2+b_1b_2+c_1c_2}
{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$$
10. Equation of a Line (Vector Form)
$\vec r = \vec a + \lambda \vec b$
$\vec a$ is position vector of a point on line
$\vec b$ is direction vector
11. Equation of Line (Cartesian Form)
$$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$$
12. Plane Equation (Vector Form)
$(\vec r-\vec a)\cdot\vec n=0$
13. Plane Equation (Cartesian Form)
$ax+by+cz+d=0$
14. Distance of a Point from a Plane
Distance of $(x_1,y_1,z_1)$ from plane $ax+by+cz+d=0$:
$$\frac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}$$
15. Angle Between Two Planes
$$\cos\theta=\frac{a_1a_2+b_1b_2+c_1c_2}
{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$$
16. Angle Between Line and Plane
$$\sin\theta=\frac{|al+bm+cn|}
{\sqrt{a^2+b^2+c^2}}$$
17. Coplanarity Condition
Four points are coplanar if:
$$\begin{vmatrix}
x_1&y_1&z_1&1\\
x_2&y_2&z_2&1\\
x_3&y_3&z_3&1\\
x_4&y_4&z_4&1
\end{vmatrix}=0$$
18. Intersection of Line and Plane
Substitute parametric line equation into plane equation
to find point of intersection.
19. Common JEE Traps
- Wrong direction ratios
- Sign errors in plane equation
- Forgetting modulus in distance formula
- Confusing angle formulas
20. Final Revision Checklist
You have mastered this chapter if you can:
- Find distance, mid-point and section points
- Form equations of lines and planes
- Find angles between lines and planes
- Solve coplanarity problems
- Handle JEE mixed questions confidently