If we have V x W = <2, 1, -1> (Cross-Product) and V ⋅ W = 4, (Dot Product) is it possible to find the angle between vectors V and W? Note that I do not actually know values for the vectors, just their products. I was under the impression that I had to know the norms of the vectors to find the angle. Is this not the case?



Yes, it is possible:

$$\cos\theta=\frac{v\cdot w}{\left\|v\right\|\left\|w\right\|}\;,\;\;\sin\theta=\frac{\left\|v\times w\right\|}{\left\|v\right\|\left\|w\right\|}\implies$$

$$\tan\theta=\frac{\left\|v\times w\right\|}{v\cdot w}$$


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