Typically, the concept of orthogonality conjurs an image of two objects existing at a right angle (90o to one another). Let’s explore this concept.
Consider two vectors u and v, defined as
Their inner product (defined as the dot product) < u | v > can be calculated as
The fact that the dot product is zero tells us that the two vectors are orthogonal. But we can calculate the dot product using another method. We can use the relationship
The only way this product can be zero (for vectors with non-zero magnitudes) is if the angle between them causes the cosine term to vanish. This happens at π/2 radians, or 90o.
This concept of orthogonality (defined by the inner product being zero) can be extended to functions in a faily simply way. We will define the inner product of two functions (f and g)being the integrall of the product of the two functions, time a third function (w - most commonly just multiplying by 1), integrated across all relevant space over which the functions are defined.
If this integral is zero, then we will say that the two functions f and g are orthogonal.