# Vectors

**Vectors** are quantities characterized by a magnitude and a direction. In two or three dimensional space, a vector is represented by an arrow that has a certain length, an initial reference point (O) and an endpoint (x,y).

Generally a **vector** (or linear array) is an ordered list of n values ( n-tuple of real numbers ). Each value is indicated with a symbol and a subscript number that indicates its position in the list (eg x_{1},x_{2}, ..., x_{n}).

The vector symbol is instead a letter with an arrow.

$$ \vec{v} = ( x_1 \ , \ x_2 \ , ... , \ x_n ) $$

Vectors are used to represent quantities that cannot be expressed by a single value.

In physics, a practical example of a vector quantity is force.

**Note**. To distinguish vectors from scalars, on Okpedia we write vectors in bold. For example **u**, **w**, **v** are vectors, instead u, v, w are scalars.

### Vector operations

The main operations between vectors are as follows:

**Vector Addition**

If (x_{1},y_{1}) and (x_{2},y_{2}) are the endpoints of two vectors**u**e**v**, the (x_{1}+x_{2},y_{1}+y_{2}) is the endpoint of the vector**u+v**.

The sum of two vectors in the R^{2}=RxR space is also obtained using the so-called**parallelogram method**.

**Scalar Multiplication**

If (x, y) is the end point of the vector v, the product kv is a vector with the end point (kx, ky).

The product kv of a real number k by vector v is also obtained by multiplying the magnitude of v by k in the same direction if k> 0 or in the opposite direction if k <0.