How to get angle bewteen two vectors in range -1 to 1 without using arc cosine?

Question

Given two normalized vectors in 3d space, how can I get a value from $-1$ to $1$ based on their angle without using arc cosine?

With use of arc cosine, I think this would give me the correct result. But since arc cosine is a computational expensive function in computer programming, I need to avoid it.

$$\frac{cos^{-1}(a * b)}{180^°}-1$$

It is acceptable that the solution doesn't return the same values as the formula above. It's just important that the output depends (not necessarily linear) on the angle and is in the range from $-1$ to $1$.

Answer 1

You could just use the dot product, since $a.b = \|a\|\|b\|cos(\angle(a,b))$. Provided your vectors are normalized, the dot product gives you $1$ if the vectors are aligned, and $-1$ if they are opposed to each other. If the vectors are not normalized, juste divide by their norm the dot product.

Answer 2

You can consider the cross product of the vectors and take the arcsin of the magnitude of that cross product. Specifically, for the case of 2D vectors $\vec{a}$ and $\vec{b}$, we have

$$\sin{\theta} = \frac{a_x\, b_y - a_y\,b_x}{\sqrt{a_x^2+a_y^2}\sqrt{b_x^2+b_y^2}}$$