Vectors: Fundamental Concepts

Definition and Properties

A vector is a mathematical object characterized by both magnitude (length) and direction. Vectors are commonly used in calculus and physics to represent quantities such as displacement, velocity, and force.

• Magnitude: The length of the vector, denoted as .

• Direction: The orientation from the initial point to the final point.

• Notation: A vector from point A to point B is written as .

Key Point: The magnitude of a vector is always positive: .

Operations with Vectors

Addition of Vectors

Vectors can be added using the parallelogram rule. The sum of two vectors and is a new vector , which represents the diagonal of the parallelogram formed by $\vec{u}$ and $\vec{v}$.

• Commutative property:

• Associative property:

Multiplication by a Scalar

Multiplying a vector by a scalar changes its magnitude but not its direction (unless $k$ is negative, which reverses the direction).

• If , points in the same direction as , with length .

• If , points in the opposite direction, with length .

Properties of Vector Operations

• There is a zero vector such that .

• For each vector , there exists such that .

• Distributive property: .

• Associative property for scalars: .

Vectors in Two Dimensions (2D)

Component Form and Unit Vectors

In 2D, vectors are represented using unit vectors and , which point in the direction of the x and y axes, respectively.

• and

• Any vector can be written as

• Vector addition:

• Scalar multiplication:

Example: Let , .

• Magnitude:

• Unit vector:

Vectors in Three Dimensions (3D)

Component Form and Unit Vectors

In 3D, vectors use unit vectors , , and for the x, y, and z axes.

• , ,

• Any vector can be written as

• Vector from to :

• Distance between points:

The Dot Product (Scalar Product)

Definition and Properties

The dot product of two vectors and is a scalar quantity defined as:

• where is the angle between the vectors.

• If and are perpendicular (), then .

• In 2D: , ,

• In 3D: , ,

• Properties: ,

Example: For and , find such that is perpendicular to :

• Set

Projections of Vectors

Definition and Calculation

The projection of vector onto vector is a vector parallel to $\vec{v}$ with length .

• Unit vector in direction of :

• Projection formula:

• Decomposition: , where is perpendicular to

Example: Express as a sum of vectors , where and :

• Check:

Vectors in n Dimensions

Generalization

Vectors can be extended to n dimensions, where a vector is an ordered set of n real numbers.

• Basis vectors: , , ...,

• Dot product:

• Magnitude:

• Angle between vectors: