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