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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.

The inner product , 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
.
This product contains partial information about the relative direction
of any two vectors, since it vanishes if they are perpendicular. In
order to capture the remaining information about direction, another
product is conventionally introduced, the vector cross product.

The cross product of two vectors is a
vector of magnitude in the direction
perpendicular to and , such that , and form a righthanded set.
Mark Ashdown