Vectors in Mathematics

Vectors

In mathematics, a vector is a quantity that possesses both magnitude and direction, but not a fixed location. Vectors virtually always signify physical entities such as displacement, velocity, acceleration, or force. They appear in various forms, including physical, geometrical, or mathematical expressions. Originally appearing towards the end of the nineteenth century, vectors became crucial in fields like physics, mechanics, electrical engineering, and other sciences due to their ability to represent forces mathematically.

Vectors can be visualized as directed line segments with the length being their magnitudes. These line segments have unique directions, allowing them to exist in different locations without changing their essence. Vectors in three-dimensional space can be described using Cartesian coordinates x, y, and z.

Operations on Vectors

Addition and Subtraction

To add two vectors, move them to the origin and complete the parallelogram formed by the two vectors. Vector addition is commutative and associative. Similarly, vector subtraction involves moving the vectors so that the tails touch and completing the parallelogram.

Cross Product(⨯)

The cross product of two vectors results in another vector that is perpendicular to both. The magnitude of the cross product is given by |v⨯w| = v * w sin θ, where θ is the smaller angle between the vectors.

Dot Product(•)

The dot product, also known as scalar product or inner product, yields a scalar value. It measures the similarity or cosine of the angle between two vectors. The dot product is given by v • w = v * w cos θ, where θ is the smaller angle between the vectors. A typical physical application of the dot product is calculating work, which can be expressed as W = Fd cos θ, where F represents a constant force acting on an object and d is the distance traveled by the object.

Unit Vectors

A unit vector, denoted by u, is a vector with a magnitude of 1. Its direction is described by its components (xu, yu, zu), which are either equal to 1, -1, or 0. For example, if xu = -1, yu = 0, and zu = 1, then we call it i or -i. In three dimensions, i, j, k would be our standard basis vectors.

Vectors have numerous applications in various areas of science and engineering, including mechanical systems and electrical circuits. They provide a powerful tool for representing complex relationships and performing calculations in these fields.

Explore the concept of vectors in mathematics, a fundamental quantity with both magnitude and direction. Learn about vector operations like addition, subtraction, cross product, and dot product. Understand the significance of unit vectors and their applications in science and engineering.