Why is theta restricted to [0,pi] in mathematics and physics?

[evinda]

Hello! (Wave)

I am looking at the following:

View attachment 5324Why do we take the $\theta$ to lie on $[0, \pi]$ ?

 

[Fantini]

Because in the interval $[0,\pi]$ the sine function $\sin(\theta)$ is nonnegative. He could have chosen $| {\mathbf a} \times {\mathbf b}| = {\color{red} (-1) } |{\mathbf a}| |{\mathbf b}| \sin(\theta)$, but then he would have reversed the orientation of the cross product.

 

[I like Serena]

Why do we take the $\theta$ to lie on $[0, \pi]$ ?

Hey evinda! (Smile)

The angle between 2 vectors is between $0$ and $\pi$.
That is, if they coincide the angle is $0$, and if they are opposite the angle is $\pi$. (Nerd)

 

[evinda]

Hey evinda! (Smile)

The angle between 2 vectors is between $0$ and $\pi$.
That is, if they coincide the angle is $0$, and if they are opposite the angle is $\pi$. (Nerd)

So it couldn't be for example $\frac{5 \pi}{4}$? Why? (Thinking)

 

[I like Serena]

So it couldn't be for example $\frac{5 \pi}{4}$? Why? (Thinking)

If the angle between 2 vectors is $\frac{5 \pi}{4}$, that's effectively the same as an angle of $\frac{3 \pi}{4}$.
Conventionally, the angle between any 2 entities is defined to be the smallest that is applicable. (Nerd)

 

[Fantini]

Perhaps this less-than-artistic picture will help.

View attachment 5325

When we consider angles between lines (and vectors) we always assume the angle that is less than or equal to $\pi$. The picture illustrates the scenario you are considering. In this case what we call the angle between the vectors is the black angle, which is equal to $\frac{3 \pi}{4}$, and not the red angle, which is equal to $\frac{5 \pi}{4}$.

Same reason why when you have two lines forming an acute angle, such as $30^{\circ}$, you say the angle between them is that and not $330^{\circ}$.

 

[evinda]

Perhaps this less-than-artistic picture will help.
When we consider angles between lines (and vectors) we always assume the angle that is less than or equal to $\pi$. The picture illustrates the scenario you are considering. In this case what we call the angle between the vectors is the black angle, which is equal to $\frac{3 \pi}{4}$, and not the red angle, which is equal to $\frac{5 \pi}{4}$.

Same reason why when you have two lines forming an acute angle, such as $30^{\circ}$, you say the angle between them is that and not $330^{\circ}$.

So is there something like a period?

 

[I like Serena]

So is there something like a period?

Erm... sure... an angle of $\frac{3\pi}4$ is equivalent to an angle of $\frac{11\pi}4$.
But as an angle between vectors we'll still refer to it as $\frac{3\pi}4$. (Thinking)

 

[Deveno]

In mathematics, the angles between $0$ and $2\pi$ are all considered *distinct*.

In physics, one often speaks of the "angle between", which is "directionless".

Often, a middle path is to use (mathematical) angles on the interval $(-\pi,\pi]$, and then to see the physical angle as the ABSOLUTE VALUE of the mathematical angle, just as speed is the absolute value of velocity.

Put another way, there is some ambiguity as angle as a function of two RAYS, and angle as a function of two LINES. Part of this has to do with the ambiguities inherent in choosing an orientation (clockwise versus counter-clockwise, in the plane, for example, as being "positive rotation"). Spaces don't come with an orientation, only our descriptions of them do. Thus, in the "physical world" the assignment of SIGN to certain quantities is, in effect, arbitrary. Sometimes it is good to define quantities in such a way as to avoid such niggling questions.