Vectors and three-dimensional geometry from Class 12: position vectors and section formulas, products of vectors, direction cosines, and the equations of lines and planes in space, a regular part of JEE Main. Start with the vector pages, since the 3D geometry pages build on them.
10 reference pages
Operations on vectors: addition, scalar multiples, the negative of a vector, dot and cross products with a worked numeric example, two classic identities, and common mistakes.
The unit vector formula in component form, unit vectors between two points, along a sum and perpendicular to two vectors, the link to direction cosines, and common mistakes.
Projection of a vector: the scalar and vector projection formulas from the dot product, special angles, a fully worked 3D example with a perpendicular check, and common mistakes.
Position vectors and head-minus-tail, the section formula internally and externally, midpoints, centroids and collinearity, with worked examples.
Direction cosines and direction ratios: converting between them, the identity l² + m² + n² = 1, worked examples from axes, two points, angles and the angle between lines, and common mistakes.
The equation of a line in 3D in vector and Cartesian form: through two points or a point and a direction, checking whether a point lies on a line, the angle between two lines, and common mistakes.
Every standard form of a plane's equation, finding a plane from a point and normal or three points, intercept form, planes through an intersection, and point-to-plane distance.
The angle between a line and a plane, two lines, and two planes, plus the parallel and perpendicular tests, with worked examples and common mistakes.
The shortest distance between skew lines and between parallel lines in vector form, how to tell intersecting lines apart, the Cartesian determinant form, and three worked examples.
Area of a triangle and parallelogram using vectors: the cross product formulas, worked examples from vertices, sides and diagonals, a check against the determinant formula, and common mistakes.