Vectors

Vectors are mathematical entities characterized by both magnitude and direction. Unlike scalars, which are defined solely by magnitude, vectors represent quantities like displacement, velocity, and force, where direction is essential. Vectors are graphically represented as directed line segments, extending from an initial point to a terminal point, denoted with bold letters or arrows placed above the symbol. Two vectors are deemed equal if they share identical magnitudes and directions, regardless of their initial positions.

Adding vectors involves placing the initial point of one vector at the terminal point of another. Alternatively, if the vectors start from the same point, their sum is the diagonal of the parallelogram formed. Subtracting vectors involves adding the negative of the second vector. Geometrically, the difference forms the opposite diagonal of the parallelogram.

Multiplying a vector by a scalar affects its magnitude, stretching, or shrinking. A negative scalar reverses the vector's direction. For instance, if vector v is multiplied by -2, the resultant vector points in the opposite direction and is twice as long.

Vectors are represented as ordered pairs of their horizontal and vertical components. If a vector v is represented in the plane with initial point A〈x₁, y₁⟩ and terminal point B〈x₂, y₂⟩, then the vector v can be written as:

The magnitude of a vector v = 〈a, b⟩ is derived using the Pythagorean theorem:

Vectors can also be expressed as linear combinations of unit vectors. A unit vector is a vector of length one, typically denoted as i for the horizontal component and j for the vertical component. A vector v can be written as:

Here, a​ and b are the horizontal and vertical components, respectively. These components can also be derived from the vector's magnitude and direction angle θ:

This representation simplifies calculations involving vector addition, subtraction, and scalar multiplication, especially when dealing with vector components in the Cartesian plane.