Why can't an inner product give $ \vec u \cdot \vec v = \|\vec u\|\|\vec v\|\cos(\theta/5) $

Question

Starting from the usual inner product axioms,

1. Linear: $(a\vec u + b\vec v)\cdot \vec w = (a\vec u \cdot \vec w) + (b \vec v \cdot \vec w)$
2. Symmetric: $\vec u \cdot \vec v = \vec v \cdot \vec u$
3. Positive definite: $\vec u \cdot \vec u \geq 0$ and is equal to zero when $\vec u = \vec 0$

Due to the Cauchy-Schwartz inequality, we know that

$$ -1 \leq \frac{\vec u \cdot \vec v}{\|\vec u\|\|\vec v\|} \leq 1$$

so it is safe define the angle $\theta$ between $\vec u$ and $\vec v$ using

$$ \theta := \cos^{-1} \left(\frac{\vec u \cdot \vec v}{\|\vec u\|\|\vec v\|} \right)$$

The inner product axioms (via the Cauchy-Schwartz inequality) guarantee this to be well-defined because the argument to $\cos^{-1}$ is between -1 and 1.

But why couldn't we say something like:

$$ \theta := 5\cos^{-1} \left(\frac{\vec u \cdot \vec v}{\|\vec u\|\|\vec v\|} \right)$$

As far as I can see, this doesn't violate any of the inner product axioms, but it breaks the link between our intuition of vectors as "oriented lengths".

Is there an ironclad reason (other than not wishing to go against geometric intuition) that the angle between vectors must be defined as $\cos^{-1} (\vec u \cdot \vec v\ /\ \|\vec u\|\|\vec v\|)$? Or, is it instead the case that there is no unique definition of the angle?

Answer 1

Let me give you a simple reason why we define the angle between the vectors $u$ and $v$ as $\theta = \cos^{-1} \left(\frac{u \cdot v }{\lVert u \rVert \lVert v \rVert}\right)$.

When we have defined an inner product in a vector space, we also have a notion of orthogonality: we say that $u$ and $v$ are orthogonal iff $\langle u,v\rangle =0$.

Substituting this in the formula for $\theta$, we get that the angle between orthogonal points $u$ and $v$ is $\pi/2$, so this really looks like a generalization of orthogonality.