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.