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A vector is a type of data that has both a magnitude (or length) and direction. There are vectors for every dimension of space (2D, 3D, etc).
Vectors are written in coordinates. The most common in the Cartesian XYZ coordinates you're used to, but there are others, such as the polar(2D), cylindrical(3D) or spherical(3D) coordinate systems.
Vectors are represented graphically as an arrow.
Vector vs scalar
Scalars are the normal numbers you're used to, like "7". Scalars are used to represent values without direction(i.e. age, luminescence, radius).
Vectors can be confusing, because in graphics they are used to represent completely different ideas. A vector can be used to represent a location in space (i.e. vertex, UV coordinate), OR it can purely represent direction (i.e. normal, tangent). Sometimes even colors are referred to as vectors, even though they do not technically fit the definition.
The magnitude of a vector is it's "length". For a location in space, the magnitude would be the distance from the origin (or it's pivot point). Some vectors do not have a magnitude (or more accurately, they always have a magnitude of 1), such as normals. These vectors are "normalized".
Types of Vectors
Points & Locations
Points in space, such as Vertices, UV points, pivots points, and object locations can all be represented with vectors. These vectors usually start at the origin, and "point" to the location of the object.
The zero vector (0,0,0). This is the "center" of your coordinate system. In "world space", this is the same for every object, and is usually represented in 3D programs with a grid (the origin is at the center of the grid).