A linear space is one upon which addition and scalar multiplication
are defined. Although such a space is often called a `vector
space', our use of the term `vector' will be reserved for the
geometric concept of a directed line segment. We still require
linearity, so that for any vectors 
and 
we must be able to
define their vector sum 
. Consistent with our purpose, we
will restrict scalars to be real numbers, and define the product of a
scalar 
and a vector 
as 
. We would like
this to have the geometrical interpretation of being a vector
`parallel' to 
and of `magnitude' 
times the
`magnitude' of 
. To express algebraically the geometric idea
of magnitude, we require that an inner product be defined for
vectors.
, also known as the dot or scalar
product, of two vectors 
and 
, is a scalar with
magnitude 
, where 
and
are the lengths of 
and 
, and 
is the angle
between them. Here 
, so that the
expression for 
is effectively an algebraic definition of
.
of two vectors is a
vector of magnitude 
in the direction
perpendicular to 
and 
, such that 
, 
and 
form a right-handed set.