Engineering Analysis/Vector Basics
Scalar Product[edit | edit source]
A scalar product is a special type of operation that acts on two vectors, and returns a scalar result. Scalar products are denoted as an ordered pair between angle-brackets: <x,y>. A scalar product between vectors must satisfy the following four rules:
- , only if x = 0
If an operation satisfies all these requirements, then it is a scalar product.
Examples[edit | edit source]
One of the most common scalar products is the dot product, that is discussed commonly in Linear Algebra
Norm[edit | edit source]
The norm is an important scalar quantity that indicates the magnitude of the vector. Norms of a vector are typically denoted as . To be a norm, an operation must satisfy the following four conditions:
- only if x = 0.
A vector is called normal if it's norm is 1. A normal vector is sometimes also referred to as a unit vector. Both notations will be used in this book. To make a vector normal, but keep it pointing in the same direction, we can divide the vector by its norm:
Examples[edit | edit source]
One of the most common norms is the cartesian norm, that is defined as the square-root of the sum of the squares:
Unit Vector[edit | edit source]
A vector is said to be a unit vector if the norm of that vector is 1.
Orthogonality[edit | edit source]
Two vectors x and y are said to be orthogonal if the scalar product of the two is equal to zero:
Two vectors are said to be orthonormal if their scalar product is zero, and both vectors are unit vectors.
Cauchy-Schwarz Inequality[edit | edit source]
The Cauchy-Schwarz inequality is an important result, and relates the norm of a vector to the scalar product:
Metric (Distance)[edit | edit source]
The distance between two vectors in the vector space V, called the metric of the two vectors, is denoted by d(x, y). A metric operation must satisfy the following four conditions:
- only if x = y
Examples[edit | edit source]
A common form of metric is the distance between points a and b in the cartesian plane: