Vectors & Angles: Find Relationship between \alpha & \beta

[Apteronotus]

Hi,

Suppose you have three vectors a, b and c.
Say the angle between a and c is given by $$\alpha$$, and between b and c by $$\beta$$.

Can we find a relationship between $$\alpha$$ and $$\beta$$?

Thanks in advance,

 

[whybother]

Well, if you know dot products then you know:

$$a . c = ||a|| ||c|| cos(\alpha)$$
$$b . c = ||b|| ||c|| cos(\beta)$$

So you could rearrange that to find a relationship between $$\alpha$$ and $$\beta$$.

 

[Apteronotus]

Thank you Whybother,

Of course you are correct. But I'm wondering can a relation be found that does not involve the dot product? Perhaps if we think of the three vectors as the edge of a parallelepiped?

 

[whybother]

Thank you Whybother,

Of course you are correct. But I'm wondering can a relation be found that does not involve the dot product? Perhaps if we think of the three vectors as the edge of a parallelepiped?

Even if you are defining a parallelepiped in 3space, I don't think you can escape from the notion of dot and cross products. Looking at a http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Parallelepiped_volume.svg/780px-Parallelepiped_volume.svg.png" of it, it seems unavoidable to me.