Vectors

The two operations are addition of two vectors and multiplication of a vector by a scalar.

Coordinate free vector algebra:

Associativity property of vector addition: (a + b) + c = a + (b + c).

1st distributive property of addition: 2(a+b) = 2a + b.

2nd distributive property of addition: (2 + 3)a = 2a + 3a.

These operations satisfy the axioms of vector spaces.

Vector decomposition:

In a given cartesian coordinate system, vectors are added coordinate-wise: