First of all, we can rewrite the inequality as
Since the inner product is positive definite, we know that this quantity will be positive. And so we can take its square root to find
This range is exactly that of the cosine function. Let’s consider the cosine restricted to the interval , where it’s injective. Here we can define an inverse function, the “arccosine”. Using the geometric view on the cosine, the inverse takes a value between and and considers the point with that -coordinate on the upper half of the unit circle. The arccosine is then the angle made between the positive -axis and the ray through this point, as a number between and .
So let’s take this arccosine function and apply it to the value above. We define the angle between vectors and by
Some immediate consequences show that this definition makes sense. First of all, what’s the angle between and itself? We find
and so . A vector makes no angle with itself. Secondly, what if we take two vectors from an orthonormal basis ? We calculate
If we pick the same vector twice, we already know we get , but if we pick two different vectors we find that , and thus . That is, two different vectors in an orthonormal basis are perpendicular, or “orthogonal”.