Example 3: Profit

Suppose a trader buys a commodity vector

A = 40i + 60j + 100k

at the price vector

P = 3i + 2j + 4k

and then sells it at the new price vector

Q = 2i + 5j + 3k.

Find his profit (or loss).

Since the trader pays P · A and receives Q · A, his profit is given by

profit = Q · A - P · A.

Thus

profit = (2 · 40 + 5 · 60 + 3 · 100) - (3 · 40 + 2 · 60 + 4 · 100) = 40.

A positive number indicates a profit and a negative number indicates a loss.