In the strictly mathematical definition of a vector, the only operations that vectors are required to possess are those of addition and scalar multiplication. (Compare this with the operations allowed on ordinary real numbers, or scalars, in which we are given addition, subtraction, multiplication, and division). For instance, in a raw vector space there is no obvious way to multiply two vectors together to get a third vector--even though we will define a couple of ways of performing vector multiplication in Vector Multiplication.

It makes sense, then, to begin studying vectors with an investigation of the operations of vector addition and scalar multiplication. This section will be entirely devoted to explaining addition and scalar multiplication of two- and three-dimensional vectors. This explanation will involve two different, yet equivalent, methods: the component method and the graphical method.