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## How to Multiply Vectors

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 right-handed set.

Mark Ashdown