Engineering Analysis/Vector Basics

From Wikibooks, open books for an open world
Jump to: navigation, search

Scalar Product[edit]

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:

  1. , only if x = 0

If an operation satisifes all these requirements, then it is a scalar product.

Examples[edit]

One of the most common scalar products is the dot product, that is discussed commonly in Linear Algebra

Norm[edit]

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:

  1. 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]

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]

A vector is said to be a unit vector if the norm of that vector is 1.

Orthogonality[edit]

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-Schwartz Inequality[edit]

The cauchy-schwartz inequality is an important result, and relates the norm of a vector to the scalar product:

Metric (Distance)[edit]

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:

  1. only if x = y

Examples[edit]

A common form of metric is the distance between points a and b in the cartesian plane: