Three-dimensional geometry is one of the highest-weighted topics in JEE Main (3-4 questions annually), covering direction cosines and ratios, equations of lines and planes, distances, angles, and intersections.

The coordinate system uses three mutually perpendicular axes. The distance between (x1,y1,z1) and (x2,y2,z2) is sqrt((x2-x1)^2+(y2-y1)^2+(z2-z1)^2). The section formula divides the join of two points in ratio m:n at $\frac{(mx2+nx1}{(m+n)}$, $\frac{my2+ny1}{(m+n)}$, $\frac{mz2+nz1}{(m+n)}$).

Direction cosines (l,m,n) of a line are the cosines of angles with coordinate axes, satisfying l^{2+m}^{2+n}^2=1. Direction ratios (a,b,c) are proportional to DCs: l=$\frac{a}{sqrt}$(a^{2+b}^{2+c}^2), etc. Two lines with DRs (a1,b1,c1) and (a2,b2,c2) are perpendicular if a1a2+b1b2+c1*c2=0, parallel if a1/a2=$\frac{b1}{b2}$=$\frac{c1}{c2}$.

A line through (x1,y1,z1) with DRs (a,b,c) in symmetric form: $\frac{x-x1}{a}$=$\frac{y-y1}{b}$=$\frac{z-z1}{c}$. Vector form: r=a+t*b. A plane ax+by+cz=d has normal (a,b,c). Intercept form: x/a+y/b+z/c=1. Normal form: lx+my+nz=p where (l,m,n) is the unit normal and p is distance from origin.

The angle between two planes is the angle between their normals: cos(theta)=|n1.n2|/(|n1||n2|). The angle between a line and a plane satisfies sin(theta)=|b.n|/(|b||n|) where b is the line's direction.

Distance from point (x0,y0,z0) to plane ax+by+cz=d is |ax0+by0+cz0-d|/sqrt(a^{2+b}^{2+c}^2). Distance between parallel planes ax+by+cz=d1 and ax+by+cz=d2 is |d1-d2|/sqrt(a^{2+b}^{2+c}^2).