Angle Between Two Vectors Discrepancy

Question

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?

Answer 1

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.

Answer 2

The angle between two vectors should always be a positive number between $0$ and $\pi$ inclusive, so if the cosine of the angle is a negative number, the value should be between $\pi/2$ and $\pi$ inclusive.

Your numbers $-2/\sqrt5$ and $1/\sqrt5$ clearly obey $\cos^2\theta+\sin^2\theta=1$ and thus form a valid sine and cosine pair, so the value for the $\sin^{-1}$ should be $\pi-\theta$.

Physically, the cross product gives the amount to which vector $\mathbf{w}$ is perpendicular to $\mathbf{v}$. This is ambiguous in that four different vectors, $\mathbf{w}$, $-\mathbf{w}$, and two other vectors of the same length but the direction $\pi-\theta$ with respect to $v$, $\mathbf{x}$ or $-\mathbf{x}$, could all have identical lengths, and have the same absolute value of cross product as $\mathbf{v}\times\mathbf{w}$. See image.