Vector Algebra and Calculus Concepts

Flashcards focused on essential vocabulary and definitions related to vector algebra, calculus, and coordinate systems.

Vector Addition

The process of combining two or more vectors to find a resultant vector.

Position Vector

A vector that represents the position of a point in space relative to an origin.

Distance Vector

A vector that points from one position to another, representing the separation between two points.

Dot Product

A scalar product of two vectors that equals the product of their magnitudes and the cosine of the angle between them.

Cross Product

A vector product of two vectors that results in a third vector perpendicular to the plane formed by the original vectors.

Magnitude of a Vector

The length or size of a vector calculated using the square root of the sum of the squares of its components.

Unit Vector

A vector with a magnitude of one, indicating direction only.

Commutative Property

A property stating that the order of addition or multiplication does not affect the result (i.e., A + B = B + A).

Distributive Property

A property that allows us to multiply a single term by two or more terms inside a set of parentheses (i.e., A(B + C) = AB + AC).

Parallelogram Rule

A method for adding vectors by forming a parallelogram where the diagonal represents the resultant vector.

Head-to-Tail Rule

A method of vector addition whereby the tail of one vector is placed at the head of the previous vector to find the resultant.

Scalar Field

A function that assigns a scalar value to every point in a space.

Vector Field

A function that assigns a vector to every point in a space.

Gradient Operator

A vector operator that represents the rate and direction of change in a scalar field.

Divergence

A scalar operator that measures the magnitude of a source or sink at a given point in a vector field.

Curl

A vector operator that describes the rotation of a vector field around a point.

Laplacian

An operator that measures the rate at which a quantity changes at a point, defined as the divergence of the gradient.