Vector

Overview
A single vector is represented by an arrow which is usually in 2D or 3D space. Vectors can be defined by either euclidean coordinates (x,y,z) or angle & magnitude based coordinates (direction, pitch, magnitude). Angle-based coordinates in 2-Dimensions always take the name "polar" coordinates, and "cylindrical" coordinates have the same principle, however they suppose that the plane that these coordinates are lying on is also itself in a 3D position, in other words, cylindrical coordinates include the Z component. When introducing a new angle in 3D angle-based coordinates; pitch, you are given the ability to define how "high" the vector is pointing; thus, we have spherical coordinates, which can define any point on a sphere of any size.

Scalars
A scalar is a "normal number" which doesn't have a position. For example, a quantity (how much of something) and a magnitude (size/power of something) are both different scalars. Even when speaking about 1-dimensional space.

Unit Vectors
"Unit" vectors, "Normal" vectors, or "Direction-only" vectors always have a length of one. The purpose of this is so that they their direction can be compared or multiplied onto another vector.

Vector Operations
Vector operations can either return a scalar (a quantity/magnitude) or another vector. To return another vector you want to preform the operation for each individual component; however, to return a scalar you usually want to add all of these operations together. For example:

new_vector_x = ax+bx new_vector_y = ay+by new_vector_z = az+bz new_vector_scalar = (ax+bx) + (ay+by) + (az+bz)

The parentheses are used to help show the X, Y and Z components.