Inner Product

In the preceding sections we studied the sum of two vectors and the product of a scalar and a vector. We shall now define the inner product (or scalar or dot product) of two vectors A and B, denoted by A · B.

The inner product arises in quite different ways in physics and economics. We first discuss an example from economics.

If the price per unit of a commodity is p, the cost of a units of the commodity is the product pa. Similarly, if a pair of commodities has price vector

P = p₁i + p₂j,

the cost of a commodity vector

A = a₁i + a₂j

is found by adding the products of the prices and quantities,

cost = p₁a₁ + p₂a₂.

If three commodities have price vector

P = p₁i + p₂j + p₃k,

the cost of a commodity vector

A = a₁i + a₂j + a₃k

is the sum of products,

cost = p₁a₁ + p₂a₂ + p₃a₃.

Notice that the cost is always a scalar. The quantity

p₁a₁ + p₂a₂ + p₃a₃

is the inner product of the vectors P and A.