Find the sine sign given a pair of 3D vectors

Question

I want to find the exact sine between two vectors in 3-dimensional space.

Data:

• $x$: vector
• $y$: vector
• $z = \Vert x \times y \Vert$

I have tried this:

$$\sin \alpha = \frac{\Vert z\Vert}{( \Vert x\Vert\cdot \Vert y\Vert)}$$

but i obtain only the absolute value of the angle, is there a way to obtain even the sign?

Answer 1

But if you know a little about inner product then you have that the angle $\,\theta\,$ between both vectors fulfills:

$$\cos\theta=\frac{x\cdot y}{||x||\;||y||}$$

and now simply use the trigonometric identity

$$\sin x=\sqrt{1-\cos^2x}$$

to get what you want. (Warning: the sign above must be carefully chosen depending on the wanted/given domain of $\,\theta\,$ )

Answer 2

Hint: Take the projection of $x$ onto $y$, and check if they are pointing in the same, or opposite directions.

Note that your given formula for $\sin \theta$ is incorrect. It should involve $||z||$.