Have a question about the angle related to cross product and dot product as it relates to to the unknown angle between two vectors.

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- chrisplusian

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- 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

dot product is ok.
they are just saying they want angle x in
|dw:1442182565780:dw|

- chrisplusian

The question literally says Given vectors A and B determine the minimum angle between A and B

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- IrishBoy123

0

- IrishBoy123

or \(- \infty\)

- 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

post or link the question,

- chrisplusian

So the question gives specific values for A and B I am just asking about the concept

- chrisplusian

Ok substitute A =-3i + j -2K, an B = 2i - 5j + k into the above question and that is what it says verbatim

- 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

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

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

**my textbook says that to find the "minimum" angle you use cross product**
nonsense

- IrishBoy123

:p

- chrisplusian

@IrishBoy123 that was my question.... is that nonsense?

- IrishBoy123

work your question for this:
A =-3i + j -2K,
B = 2i - 5j + k
your suggestion!

- 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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