Verifying Angle Between Vectors: \vec u & \vec v

In summary, the equations given state that theta is the same when the vectors are dot and cross products of each other, but when arcsin and arccos are used to solve the equations, the results are different.
  • #1
tamtam402
201
0

Homework Statement



Using these 2 vectors:

[itex] \vec u = (3,-4,0)[/itex]

[itex] \vec v = (1,1,1)[/itex]

I must verify that theta is the same with these 2 equations:

Dot product
[itex] \vec u \bullet \vec v = ||\vec u|| ||\vec v|| cos( \theta)[/itex]

Cross product
[itex] ||\vec u \wedge \vec v|| = ||\vec u|| ||\vec v|| sin( \theta)[/itex]

Homework Equations



They were given in 1)

The Attempt at a Solution



I did all the calculations, I get the following answers:

[itex] ||\vec u || = 5[/itex]

[itex] ||\vec v || = \sqrt{3}[/itex]

[itex] \vec u \bullet \vec v = -1[/itex]

[itex] ||\vec u \wedge \vec v|| =\sqrt{74}[/itex]I then solve the 2 equations given above using arcsin and arccos to find the values of theta, but I get 96.6 using the dot product, and 83.3 using the cross product. The weird thing is that 180-83.3 = 96.6...

I must be missing something obvious, but I can't understand why I get the wrong answer :(
 
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  • #2


2 vectors define a plane

note that within that plane you can consider the smaller (<=90) or larger (>=90) angle between the 2 vectors , but they will always sum to 180degrees
 
  • #3


lanedance said:
2 vectors define a plane

note that within that plane you can consider the smaller (<=90) or larger (>=90) angle between the 2 vectors , but they will always sum to 180degrees

Basically the other angle is the one between the vector and the grey line?? If so, then it can't be considered the angle between the 2 vectors. It would be right if one of the vectors was pointing into the opposite direction.

How can you tell which value is the "right" one when trying to determinate the angle between the 2 vectors using the cross product? I'm trying to visualize the vectors in my head, and I know there is only one "right" answer.

http://i.imgur.com/uM2ni.jpg
 
  • #4


Hint: Arcsin will always give you a result between -90 degrees and 90 degrees, yet the angle between two vectors ranges from 0 to 180 degrees.
 
  • #5


vela said:
Hint: Arcsin will always give you a result between -90 degrees and 90 degrees, yet the angle between two vectors ranges from 0 to 180 degrees.

Yes, but that doesn't tell me which answer is right. Arcsin could return 30 degress, and it could be the right answer, but the right answer could also be 150. How are you supposed to tell which one is right?
 
  • #6


note that the cross product expression is a magnitude
[tex]
|u \times v| = |u||v|sin(\theta) \geq 0
[/tex]

the dot product allows negative values which will occur when the angel is greater than 90 degrees

so in short, use the dot product
 
  • #7


You can't conclusively determine the angle from the arcsin alone, just as you can't tell me what x equals with certainty if all I told you is sin x = 0.5.
 
  • #8


To add onto vela's comments

Consider a plot of sin(t) with t from -pi to pi

In the region -pi to 0 , sin(t) is negative. As you are dealing with magnitudes [itex] \frac{ |u \times v|}{ |u||v|}[/itex] will never be negative, so the arcsin will only return a value in the range 0 to pi

now on a plot of 0 to pi, the graph of sin(t) is symmetric about pi/2.

So say you know sin(t) = 0.5. This could be either t=30 or t=150. The calculator will always return a number in the range (-90 to 90) so in this case 30deg.
 
  • #9


Ok I get it, thanks guys :D
 

1. What is the formula for calculating the angle between two vectors?

The formula for calculating the angle between two vectors, ν, is given by:

ν = cos-1[(ηθ) / (|η| • |θ|)]

where η and θ are the two vectors, and and | | denote the dot product and magnitude, respectively.

2. How do I verify the angle between two vectors?

To verify the angle between two vectors, you can use the dot product formula mentioned above. If the dot product is equal to zero, then the vectors are perpendicular and the angle between them is 90 degrees. If the dot product is positive, the angle is acute, and if it is negative, the angle is obtuse.

3. Can the angle between two vectors be greater than 180 degrees?

No, the angle between two vectors cannot be greater than 180 degrees. This is because the dot product formula takes into account the magnitudes of the vectors, ensuring that the angle is always between 0 and 180 degrees.

4. What is the difference between the angle between two vectors and the angle of rotation?

The angle between two vectors is the geometric angle formed between them, while the angle of rotation is the amount by which a vector is rotated about a fixed point. The angle of rotation can be calculated using trigonometric functions, while the angle between two vectors is calculated using the dot product formula.

5. How can I use the angle between two vectors in real life applications?

The angle between two vectors has various applications in physics, engineering, and computer graphics. For example, in physics, the angle between a force vector and a displacement vector can be used to calculate the work done by the force. In computer graphics, the angle between two vectors can be used to determine the direction of light and the color of a pixel on a screen. It is also used in navigation and robotics to calculate the direction and distance between two points.

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