# Vectors and Rectangular Coordinates

### Unit Vectors

Consider the Cartesian coordinate system in space. On each of its dimensions x,y and z, we define respectively the vectors:

These vectors are independent between them and have magnitude (length) of one:

We call these vectors, unit vectors. Every other vector in this space can be constructed by adding multiples of these unit vectors. Or, a vector can be decomposed in components, in multiples of unit vectors, at all dimensions. So the vector in Image 1 can be constructed by adding:

**Image 1: A Vector and its Components**

We have to note that we may have from one, up to infinite number of dimensions. We used three dimensions just to make it clearer.

### Magnitude and Direction

We will examine the magnitude and the direction of the vectors in two and three dimensions. Similarly, the examination can be extended to higher number of dimensions.

**XY Plane**

In image 2, we can see a vector in the XY plane. From trigonometry we can calculate the magnitude of the vector and its components:

**Image 2: A Vector on the XY Plane**

**XYZ Plane**

In image 3, we can see a vector in the XYZ space. Again, from trigonometry we can calculate the magnitude of the vector and its components.

**Image 3: A Vector in the XYZ Space**