1 Vectors (i) Vectors and vector products

Vector

A quantity that has both magnitude and direction, such as velocity, acceleration, and force.

Scalar

A quantity that can be specified by a single number, such as length, area, volume, mass, density, temperature, etc.

Unit vector

A vector with a magnitude equal to one, usually denoted as 𝒓̂.

Vector addition

The operation of adding two or more vectors together to form a resultant vector.

Direction cosines

The cosines of the angles that a vector makes with the coordinate axes.

Scalar product (dot product)

A product of two vectors that results in a scalar value, calculated as a.b = |a||b| cos(θ).

Vector product (cross product)

A product of two vectors that results in a vector, calculated as a × b = |a||b| sin(θ) n̂, where n̂ is perpendicular to the plane of a and b.

Components of a vector

The projections of a vector along the axes of a coordinate system.

Magnitude of a vector

The length or size of the vector, calculated using the Pythagorean theorem for its components.

Scalar triple product

The product obtained from a vector and the cross product of two other vectors, giving a scalar result.

Vector triple product

The product formed by taking the cross product of a vector with the cross product of two other vectors.

Right-handed Cartesian coordinate system

A coordinate system where the positive direction of the axes follows the right-hand rule.

Equal vectors

Two vectors that have the same magnitude and direction.

Collinear vectors

Vectors that lie on the same line or are in the same direction.

Cyclic permutation

A rearrangement of elements in a circular fashion, used in vector triple products.