Confusion over angle between vectors

[ThomasMagnus]

Homework Statement

I have been doing dot and cross product recently. I get how to calculate everything; however, I am confused about which angle to use when asked to find the angle between two vectors. When you use the cross product, you always end up with 2 answers, for example 120° and 60°. However, if you were to find the angle between them using dot product, you would only get one unique answer, as cos 120 and cos 60 are the opposite sign. So my question is, when asked to find the angle between two vectors given in i j k notation, is it best to always do it using dot product to avoid this confusion?

Also, is it correct to say that the angle found between vectors using the dot product MUST be the same as when using the cross product method?

Homework Equations

$\vec{a}$$\bullet$$\vec{b}$=|a||b| cos $\theta$

$\vec{a}$ $\times$ $\vec{b}$=|a||b| sin $\theta$

The Attempt at a Solution

N/A

 

[rude man]

When in doubt, revert to basics:

a = ax i + ay j
b = bx i + by j
a dot b = ax bx + ay by since i dot i = j dot j = 1
a x b = set up the determinant:
1st row: i j k
2nd row: ax ay 0
3rd row bx by 0

 

[ThomasMagnus]

When in doubt, revert to basics:

a = ax i + ay j
b = bx i + by j
a dot b = ax bx + ay by since i dot i = j dot j = 1
a x b = set up the determinant:
1st row: i j k
2nd row: ax ay 0
3rd row bx by 0

Thing is, I never do it that way really. I guess since arccosine is defined between 0,pi it will have to give the unique and only possible angle?

Thanks :)

 

[rude man]

Thing is, I never do it that way really. I guess since arccosine is defined between 0,pi it will have to give the unique and only possible angle?

Thanks :)

I understand.

I believe the instructions say "the angle between the directions of the vectors". So if you pick the SMALLER angle between the two vectors you will get the right answer. So that angle never exceeds 180 deg.

For the cross-product you still also need to understand and remember the right-hand rule to get the direction of the cross-product vector. That vector will always be perpendicular to both a and b.

 

[ThomasMagnus]

I understand.

I believe the instructions say "the angle between the directions of the vectors". So if you pick the SMALLER angle between the two vectors you will get the right answer. So that angle never exceeds 180 deg.

For the cross-product you still also need to understand and remember the right-hand rule to get the direction of the cross-product vector. That vector will always be perpendicular to both a and b.

Say you use dot product and get 109 degrees, and using cross product you get 71 or 109. It would have to be 109 right?

 

[rude man]

Say you use dot product and get 109 degrees, and using cross product you get 71 or 109. It would have to be 109 right?

I don't quite understand your question.

The "angle between the directions" has nothing to do with whether you're finding the dot or the cross product. It's just the angle between the two vectors.

For dot product the answer is |a|*|b| cosθ and for the cross product the answer is |a|*|b| sinθ with direction determined by the right-hand rule.

You don't "get" an angle to determine either product.