chrisplusian
  • chrisplusian
Have a question about the angle related to cross product and dot product as it relates to to the unknown angle between two vectors.
Mathematics
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katieb
  • katieb
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chrisplusian
  • chrisplusian
When a question asks you to find "the minimum angle between" two vectors, is it ok to use the dot product definition? Or do you have to use cross product to find the "minimum' angle?
phi
  • phi
dot product is ok. they are just saying they want angle x in |dw:1442182565780:dw|
chrisplusian
  • chrisplusian
The question literally says Given vectors A and B determine the minimum angle between A and B

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IrishBoy123
  • IrishBoy123
0
IrishBoy123
  • IrishBoy123
or \(- \infty\)
chrisplusian
  • chrisplusian
So i did it with the dot product and got the larger of the two angles, and AI was wondering if I had to use the cross product to ensure I get the smaller of the two? It has been about two years since I have done any three dimension vector operations and my mind is fuzzy when it comes to the topic
IrishBoy123
  • IrishBoy123
post or link the question,
chrisplusian
  • chrisplusian
So the question gives specific values for A and B I am just asking about the concept
chrisplusian
  • chrisplusian
Ok substitute A =-3i + j -2K, an B = 2i - 5j + k into the above question and that is what it says verbatim
chrisplusian
  • chrisplusian
But I am not looking for help with this specific question, I am trying to find out the concept in general, because my textbook says that to find the "minimum" angle you use cross product. That seemed sketchy to me so I wanted to inquire about the concept
IrishBoy123
  • IrishBoy123
these are 2 planes, and they are defined by their normals. the angles between those normals are fixed. you can cross them or dot them or whatever, just remember the right hand rule.
phi
  • phi
I think the are saying |dw:1442183770146:dw| if you use dot product and get an angle bigger than 90ยบ, do 180-x to find the "minimum angle"
IrishBoy123
  • IrishBoy123
**my textbook says that to find the "minimum" angle you use cross product** nonsense
IrishBoy123
  • IrishBoy123
:p
chrisplusian
  • chrisplusian
@IrishBoy123 that was my question.... is that nonsense?
IrishBoy123
  • IrishBoy123
work your question for this: A =-3i + j -2K, B = 2i - 5j + k your suggestion!
IrishBoy123
  • IrishBoy123
the cross product of A & B will give you a third vector that is at right angles to both of these according to the definition \(A \times B = |A||B|\sin\theta \ \hat n\). importantly, the magnitude of that vector will equal \(|A||B|\sin\theta\) where \(\theta \) is the angle between the vectors. so \(sin \theta = \frac{|A \times B|}{|A| \ |B|}{}\) so you can do it that way, find the cross product (which will be a vector), then plug its *magnitude* into the equation the dot product on the other hand will give you just a number which equals the projection of either vector onto the other. and it's simpler, chore-wise again we have the definition \(A \bullet B = |A||B| \cos \theta \) so \(cos \theta = \frac{A \bullet B}{|A| \ |B|}{}\) you may then run into the problem as indicated above |dw:1442224407955:dw| i did when i cranked out the numbers that's a bit of a mouthful so i hope it is helpful :p

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