# Engineering Analysis/Vector Basics

## Scalar Product

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. $\langle x,x\rangle \geq 0,\quad \forall x\in V$
2. $\langle x,x\rangle =0$ , only if x = 0
3. $\langle x,y\rangle =\langle y,x\rangle$
4. $\langle x,cy_{1}+dy_{2}\rangle =c\langle x,y_{1}\rangle +d\langle x,y_{2}\rangle$

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

### Examples

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

## Norm

The norm is an important scalar quantity that indicates the magnitude of the vector. Norms of a vector are typically denoted as $\|x\|$ . To be a norm, an operation must satisfy the following four conditions:

1. $\|x\|\geq 0$
2. $\|x\|=0$  only if x = 0.
3. $\|cx\|=|c|\|x\|$
4. $\|x+y\|\leq \|x\|+\|y\|$

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:

${\bar {x}}={\frac {x}{\|x\|}}$

### Examples

One of the most common norms is the cartesian norm, that is defined as the square-root of the sum of the squares:

$\|x\|={\sqrt {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}}$

### Unit Vector

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

## Orthogonality

Two vectors x and y are said to be orthogonal if the scalar product of the two is equal to zero:

$\langle x,y\rangle =0$

Two vectors are said to be orthonormal if their scalar product is zero, and both vectors are unit vectors.

## Cauchy-Schwartz Inequality

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

$|\langle x,y\rangle |\leq \|x\|\|y\|$

## Metric (Distance)

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. $d(x,y)\geq 0$
2. $d(x,y)=0$  only if x = y
3. $d(x,y)=d(y,x)$
4. $d(x,y)\leq d(x,z)+d(z,y)$

### Examples

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

$d(a,b)_{cartesian}={\sqrt {(x_{a}-x_{b})^{2}+(y_{a}-y_{b})^{2}}}$