Vectors and Angles

  • #1
Apteronotus
202
0
Hi,

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

Can we find a relationship between [tex]\alpha[/tex] and [tex]\beta[/tex]?

Thanks in advance,
 

Answers and Replies

  • #2
whybother
166
0
Well, if you know dot products then you know:

[tex]a . c = ||a|| ||c|| cos(\alpha)[/tex]
[tex]b . c = ||b|| ||c|| cos(\beta)[/tex]

So you could rearrange that to find a relationship between [tex]\alpha[/tex] and [tex]\beta[/tex].
 
  • #3
Apteronotus
202
0
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?
 
  • #4
whybother
166
0
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" [Broken] of it, it seems unavoidable to me.
 
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