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 satisfies 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-Schwarz Inequality

The Cauchy-Schwarz 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:

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