Vectors

Vectors: Basic Concepts

  • Across various branches of mathematics, a vector is generally defined as an object possessing both magnitude (size) and direction.
  • Vectors can represent a wide variety of physical quantities such as force, velocity, or displacement in space.
  • A vector is generally represented as an arrow with a certain length and direction or as a coordinate in n-dimensional space.
  • Vectors can be added or subtracted, multiplied by scalars and have their magnitude and direction calculated.

Vector Notation

  • Usually, vectors are denoted in bold, lowercase letters such as v.
  • Vectors can also be represented in column form such as (x, y), where x and y are the components of the vector, showing how much the vector moves in each dimension.
  • Another way to denote a vector is to specify its initial and terminal points e.g., AB, where A is the initial point, and B is the terminal point.

Magnitude of a Vector

  • The magnitude of a vector refers to its length or size.
  • In two-dimensional space, the magnitude of a vector v with components (x, y) is calculated using Pythagoras’ theorem: v = sqrt(x² + y²).

Vector Operations

  • Vector addition is the operation of adding two or more vectors together to yield a resultant vector.
  • Vector subtraction involves finding the difference between two vectors, which results in another vector.
  • Scalar multiplication is the operation of multiplying a vector by a scalar (real number), resulting in a vector that is the same direction but a different magnitude.

Applications

  • Understanding vectors is crucial for many areas of mathematics and physical sciences.
  • In physics, vectors are used to describe force, velocity, and acceleration. In computer science, they’re used in computer graphics, machine learning, and many other areas.
  • In geometry, one application is finding the position vector, which describes the position of a point in space relative to the origin.
  • Vectors can also be used to find the centroid of a triangle, to calculate areas, and in many other geometric applications.

Dot Product

  • The dot product is a type of multiplication for vectors where the product equals the cosine of the angle between the vectors times the product of their magnitudes.
  • The dot product is used to find the angle between vectors and to determine orthogonality (perpendicularity).

Cross Product

  • The cross product, also known as the vector product, results in a vector perpendicular (at right angles) to the two original vectors.
  • It’s used to find the area of parallelograms and triangles and to determine parallelism and orthogonality. It is also frequently used in physics to find torque and angular momentum.

Vector Projection

  • The projection of one vector onto another represents the length of the shadow of the first vector falling onto the line determined by the second vector when a light is shining from the origin perpendicular to that line.
  • It is a useful concept in many fields, including physics and engineering.