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

[tamtam402]

Homework Statement

Using these 2 vectors:

$\vec u = (3,-4,0)$

$\vec v = (1,1,1)$

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

Dot product
$\vec u \bullet \vec v = ||\vec u|| ||\vec v|| cos( \theta)$

Cross product
$||\vec u \wedge \vec v|| = ||\vec u|| ||\vec v|| sin( \theta)$

Homework Equations

They were given in 1)

The Attempt at a Solution

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

$||\vec u || = 5$

$||\vec v || = \sqrt{3}$

$\vec u \bullet \vec v = -1$

$||\vec u \wedge \vec v|| =\sqrt{74}$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 :(

 

[lanedance]

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

 

[tamtam402]

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

 

[vela]

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.

 

[tamtam402]

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?

 

[lanedance]

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

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

so in short, use the dot product

 

[vela]

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.

 

[lanedance]

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 $\frac{ |u \times v|}{ |u||v|}$ 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.

 

[tamtam402]

Ok I get it, thanks guys :D