<s>[INST] Angle Between Two Vectors Discrepancy Here is something that is bugging me: Consider the vectors
$$ \mathbf{v} = 2\mathbf{i} - 4\mathbf{j} + 4\mathbf{k} \qquad \text{ and } \qquad \mathbf{w} = 2\mathbf{j} - \mathbf{k}. $$
We have two different formulas for the angle $\theta$ between these vectors:
$$ \theta = \cos^{-1}\left(\frac{\mathbf{v} \cdot \mathbf{w}}{|\mathbf{v}||\mathbf{w}|} \right) \qquad \text{ and } \qquad \theta = \sin^{-1}\left(\frac{|\mathbf{v} \times \mathbf{w}|}{|\mathbf{v} || \mathbf{w}|} \right) .$$
Well:
$$|\mathbf{v}| = 6, \qquad |\mathbf{w}| = \sqrt{5}, \qquad \mathbf{v} \cdot \mathbf{w} = -12, \qquad \text{ and } |\mathbf{v} \times \mathbf{w}| = 6. $$
But:
$$\cos^{-1}\left(-\frac{2}{\sqrt{5}} \right) \ne \sin^{-1}\left( \frac{1}{\sqrt{5}} \right).  $$
So, what gives?
 [/INST] Both functions only give one specific inverse, but others exist. If you remove the inverse functions and convert it to a simple system of equations, you actually have a valid solution.
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