Given two vectors u and v, its inner product, written as u.v, is the sum of the products of corresponding components.
If u= <2,3,1> and v= <-2,1,3>, then u.v = 2·(-2) + 3·1 + 1·3 = 2.
The geometric meaning of the inner product of u and v is the following:
u.v = |u|.|v|cos(t)
(The inner product of u and v is equal to the product of the length of u, the length of v, and the cosine of the angle t between u and v
This is an important formula, because we know the following facts about u and v once the inner product becomes available:
- If u.v is zero, where u and v are non-zero vectors, then cos(t) must be zero and, as a result, t must be 90 degree. Therefore, u and v are perpendicular to each other.
- If u.v is equal to the product of lengths of u and v, the cosine of t is 1 and t is 0 degree. As a result, u and v are parallel to each other and point to the same direction.
- If u.v is equal to the negative product of lengths of u and v, the cosine of t is -1 and u and v are parallel to each other but point to opposite directions.