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Lattice vectors
Consider a lattice (2D) like the one shown.

We can construct vectors between any two lattice points,

These vectors are known as lattice vectors.
Now lattice vectors a & b have been constructed.

As this is a 2D lattice two basis vectors are required to describe all lattice vectors that may be constructed.

These vectors are known as lattice vectors.
a b
If a and b are such basis vectors any other vector, t, may be wriiten in terms of a & b, the basis vectors."
i.e. t= Ua + Vb where U and V are constants.
This is given the shorthand notation of t = [UV]
Note: this extends to 3D also, e.g. t= [UVW]
a b
For example consider the vector t shown.

How would we write this in terms of the basis vectors a & b?
As can be seen t can be written as 2a + b or in shorthand [21]
2 a b t