# Vector calculator

separator : space(s)

Use this calculator to do the following calculations on a vector :
- Calculate the norm of a vector
- Calculate orthogonal vector of a given vector (2D plan)
- Normalize a vector

## Euclidean Norm of a vector

The Euclidean norm of a vector \vecu of coordinates (x, y) in the 2-dimensional Euclidean space, can be defined as its length (or magnitude) and is calculated as follows :

norm(vecu) = sqrt(x^2+y^2)

The norm (or length) of a vector \vecu of coordinates (x, y, z) in the 3-dimensional Euclidean space is defined by:

norm(vecu) = sqrt(x^2+y^2+z^2)

Example: Calculate the norm of vector [[3],[2]]

norm(vecu) = sqrt(3^2+2^2) = sqrt(13)

## Orthogonal Vector

Definition

Two vectors of the n-dimensional Euclidean space are orthogonal if and only if their dot product is zero.

The following propositions are equivalent :

- \vecu _|_ \vecv

- Vectors \vecu and \vecv are orthogonal

- Their dot product is zero: \vecu . \vecv = 0

How to calculate the orthogonal vector ?

Let \vecu be a vector of coordinates (a, b) in the Euclidean plane \mathbb{R^2}. Any \vecv vector of coordinates (x, y) satisfying this equation is orthogonal to \vecu:

\vecu . \vecv = 0

a.x + b.y = 0

If b != 0 then y = -a*x/b

Therefore, all vectors of coordinates (x, -a*x/b) are orthogonal to vector (a,b) whatever x. We note that all these vectors are collinear (have the same direction).

For x = 1, we have \vecv = (1,-a/b) is an orthogonal vector to \vecu.

## Vector Normalization

Definition : Let \vecu be a non-zero vector. The normalized vector of \vecu is a vector that has the same direction than \vecu and has a norm which is equal to 1.

We note \vecv the normalized vector of \vecu, then we have,

\vecv = \vecu/norm(vecu)

Example : Normalization of the vector of coordinates (3, -4) in the Euclidean plane

We compute its norm,

\norm(vecu) = sqrt(3^2 + (-4)^2) = sqrt(25) = 5

The normalized vector of \vecu is therefore \vecv = \vecu/norm(vecu) = (3/5 , -4/5)