A vector is a quantity with both magnitude and direction, represented as a = a_{1}i + a_{2}j + a_{3}k in 3D space. The magnitude is |a| = sqrt(a_{1}^{2} + a_{2}^{2} + a_{3}^{2}), and the unit vector is a/|a|. Vector addition follows the triangle or parallelogram law, and scalar multiplication scales the magnitude while preserving (k > 0) or reversing (k < 0) direction. The position vector of a point P(x,y,z) is the vector from origin to P. Direction cosines l, m, n satisfy $l^{2}$ + $m^{2}$ + $n^{2}$ = 1 and represent the cosines of angles made with the coordinate axes. Two vectors are collinear if one is a scalar multiple of the other. The vector from A to B is AB = (position vector of B) - (position vector of A). These fundamental concepts underpin all vector operations in JEE and form the building blocks for dot product, cross product, and their applications.