Vector Algebra

Scalar and Vector Quantities

Scalars have only magnitude (e.g., mass, speed), while vectors possess both magnitude and a specific direction (e.g., velocity, force).

Position Vector

A vector that represents the position of a point P(x, y, z) relative to the origin O, expressed as xi + yj + zk.

Direction Cosines and Ratios

The cosines of the angles made by a vector with the positive x, y, and z axes, denoted by l, m, and n where l² + m² + n² = 1.

Dot Product (Scalar Product)

The product of two vectors resulting in a scalar, defined as a . b = |a||b|cos(θ), useful for finding the angle between vectors.

Cross Product (Vector Product)

The product of two vectors resulting in a third vector perpendicular to both, defined as a x b = |a||b|sin(θ)n̂, useful for finding areas.