# 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: