Basic Lin Alg - Angle Between Vectors

In summary, the conversation discusses finding the angle between two vectors using the dot product and the equation cos(theta) = <u, v> / ||u||||v||. After some initial confusion regarding a scalar versus a vector, the correct value of cos(theta) is found to be -1/2, resulting in an angle of 120 degrees or 2pi/3 radians.
  • #1
sunmaz94
42
0

Homework Statement



Find the angle between the vectors [-1, 1, 2] and [2, 1, -1].

Homework Equations



cos(theta) = <u, v> / ||u||||v||

The Attempt at a Solution



I get <u, v> = -3 and both norms equal to the square root of six. Then cos(theta) equals (-1/12), but I don't think this is correct. Can someone find my careless error? Thanks.
 
Physics news on Phys.org
  • #2
Your <u,v> must be a vector, not a scaler.
 
  • #3
[tex]
\cos\theta = \frac{-3}{\sqrt{6}\sqrt{6}} = ?
[/tex]

:P
 
  • #4
magicarpet512 said:
Your <u,v> must be a vector, not a scaler.

No...dot products always yield scalars. Thanks anyways.
 
  • #5
magicarpet512 said:
Your <u,v> must be a vector, not a scaler.

No, [itex] \langle \cdot, \cdot \rangle [/itex] is the inner product, the same thing as the dot product.

[itex] \langle \vec{u}, \vec{v} \rangle [/itex] is the same as [itex] \vec{u} \cdot \vec{v} [/itex].
 
  • #6
spamiam said:
[tex]
\cos\theta = \frac{-3}{\sqrt{6}\sqrt{6}} = ?
[/tex]

:P

I did sqrt(6)^2 = 36 instead of 6! Wow! Stupid. Thanks!
 
  • #7
spamiam said:
No, [itex] \langle \cdot, \cdot \rangle [/itex] is the inner product, the same thing as the dot product.

[itex] \langle \vec{u}, \vec{v} \rangle [/itex] is the same as [itex] \vec{u} \cdot \vec{v} [/itex].

Exactly.
 
  • #8
magicarpet512 said:
Your <u,v> must be a vector, not a scaler.

sorry, i didnt mean to say that!

As spamiam is hinting, how do you get [tex]\theta[/tex]?
 
  • #9
magicarpet512 said:
sorry, i didnt mean to say that!

As spamiam is hinting, how do you get [tex]\theta[/tex]?

I've figured it out now...careless error...cos(theta) is -1/2 so theta is 120 degrees or 2pi/3 radians.
 

1. What is the definition of the angle between two vectors?

The angle between two vectors is the measure of the deviation between the two vectors when they share a common origin point. It is represented in degrees or radians and can be calculated using geometric or trigonometric methods.

2. How is the angle between two vectors related to their dot product?

The angle between two vectors is directly related to their dot product. The dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them. In other words, the dot product of two vectors is equal to the product of their lengths and the cosine of the angle between them.

3. Can the angle between two vectors be negative?

No, the angle between two vectors cannot be negative. It is always a positive value between 0 and 180 degrees (or 0 and π radians). This is because the angle between two vectors is a measure of the deviation between them, not the direction.

4. How can the angle between two vectors be used in real-life applications?

The angle between two vectors is a fundamental concept in mathematics and is used in various fields such as physics, engineering, and computer graphics. It is used to calculate forces, determine the orientation of objects, and create 3D animations.

5. Is there a formula to calculate the angle between two vectors?

Yes, there are multiple formulas to calculate the angle between two vectors depending on the information available. Some common formulas include using the dot product, the cross product, or trigonometric functions such as cosine or arccosine.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
1
Views
1K
  • Precalculus Mathematics Homework Help
Replies
16
Views
1K
  • Precalculus Mathematics Homework Help
Replies
3
Views
2K
  • Precalculus Mathematics Homework Help
Replies
1
Views
403
Replies
9
Views
1K
  • Precalculus Mathematics Homework Help
Replies
2
Views
1K
  • Precalculus Mathematics Homework Help
Replies
3
Views
1K
  • Precalculus Mathematics Homework Help
Replies
9
Views
2K
  • Precalculus Mathematics Homework Help
Replies
18
Views
5K
Replies
1
Views
1K
Back
Top