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 x₁,x₂, ..., 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 (x₁,y₁) and (x₂,y₂) are the endpoints of two vectors u e v, the (x₁+x₂,y₁+y₂) is the endpoint of the vector u+v.
  
  The sum of two vectors in the R²=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.