Understanding the Direction of Axial Vectors in Rotational Motion

[donaldparida]

Background: I have just started studying vector algebra. I studied that physical quantities like velocity, force, momentum, etc. are vectors (My book addresses them as polar vectors/true vectors) and physical quantities like quantities like angular velocity, torque, angular momentum are also vectors (My book addresses them as axial vectors/pseudovectors).
My question/problem: I really do not understand why axial vectors do not have a "curved direction" like clockwise or anticlockwise. For example, torque has a direction which is perpendicular to the force vector and the position vector which i find completely meaningless. Why does it not have a direction like clockwise or anticlockwise (I believe that they are valid directions). I found out on browsing the Internet that this is because they are cross products of two vectors and the cross product of two vectors is a vector whose direction is perpendicular to the plane containing the vectors which are being cross multiplied.

 

[A.T.]

In 3D there is no unique clockwise direction. And it depends on the perspective whether a rotation is CW or CCW. The current convention is the simplest way to encode all the needed information in 3D.

 

[donaldparida]

@A.T. Can you please elaborate on "The current convention is the simplest way to encode all the needed information in 3D.".

 

[PeroK]

Background: I have just started studying vector algebra. I studied that physical quantities like velocity, force, momentum, etc. are vectors (My book addresses them as polar vectors/true vectors) and physical quantities like quantities like angular velocity, torque, angular momentum are also vectors (My book addresses them as axial vectors/pseudovectors).
My question/problem: I really do not understand why axial vectors do not have a "curved direction" like clockwise or anticlockwise. For example, torque has a direction which is perpendicular to the force vector and the position vector which i find completely meaningless. Why does it not have a direction like clockwise or anticlockwise (I believe that they are valid directions). I found out on browsing the Internet that this is because they are cross products of two vectors and the cross product of two vectors is a vector whose direction is perpendicular to the plane containing the vectors which are being cross multiplied.

If you have motion in 2D, then angular momentum can be represented by a "signed scalar". Positive for clockwise and negative for anticlockwise. But if you have motion in 3D, then you must accept the full vector nature of angular momentum.

Mathematically, the vector (cross) product of two vectors in the x-y plane is a vector in the z-direction. That's why you can shortcut the vector notation and just have use a signed scalar (where the z-direction is implied).

 

[A.T.]

@A.T. Can you please elaborate on "The current convention is the simplest way to encode all the needed information in 3D.".

Can you propose a simpler way to encode all information about 3D rotation?

 

[donaldparida]

@A.T. , Ok. Now i got it. Let me write what i have understood (Thanks for reading).

In 3 dimensions, the rotation of an object is possible about any of the infinite number of axes of rotation (There can be infinite number of axes of rotation in 3 dimensions) and for each axis, there is a corresponding clockwise and anticlockwise sense of rotation. Thus in 3 dimensions, to describe the rotational motion of an object and the direction of vectors related to it, it is necessary to specify both the orientation of the axis of rotation as well as the sense of rotation for that particular axis . That is why the direction of an axial vector or pseudovector is perpendicular to the plane containing the two vectors of which it is the cross product and this direction is the same as the orientation of the axis of rotation. In addition, the sense of rotation of the object about an axis can either be clockwise or anticlockwise which is indicated by - and + signs respectively before the magnitude of the axial vector. The orientation of the axis of rotation thus tells us about the direction of the axial vectors in consideration and for that specific orientation of the axis of rotation or direction of the axial vector, the sense of rotation is indicated by - and + signs depending upon whether it is clockwise or anticlockwise respectively. Right?
One more question, Since axial vectors are cross products and cross products are 3 dimensional in nature, how would the direction of them be specified in 2 dimensions or is there no need of specifying the direction since they is only one axis (or point?) of rotation in 2 dimensions or do they not exist in 2 dimensions but only 3 dimensions?

 

[vanhees71]

That's correct.

Another way to state the direction of rotation in connection with axial vectors is the "right-hand rule": point with the thumb of your right hand in direction of the vector, then the fingers give the orientation of the rotation. The most simple quantity to visualize should be angular velocity.

 

[donaldparida]

@vanhees71 , @PeroK and @A.T. , I have a doubt. In 3 dimensions, the direction of axial vectors (which are cross products of two vectors) is determined by applying the right hand rule and this direction automatically encodes the sense of rotation along with orientation of the axis of rotation. Then isn't it unnecessary to point out vectors which are related to clockwise and anticlockwise sense of rotation by using - and + signs respectively?

 

[vanhees71]

The right-hand rule tells you the direction of rotation. That's it. In 3D there's no $\pm$. In 2D one defines that rotating counter-clockwise is by a positive angle.

 

[A.T.]

Then isn't it unnecessary to point out vectors which are related to clockwise and anticlockwise sense of rotation by using - and + signs respectively?

As already explained, the clockwise concept is not useful in 3D. When you look at a clock from from the back, the pointers of go anticlockwise.

For every axis there are two possible axial vector directions, corresponding to the two possible rotation directions around that axis.

 

[vanhees71]

Yes, and this ambiguity is removed by defining the orientation by the right-hand rule, nicely depicted in your figure.

 

[donaldparida]

Thanks to all! Just for the sake of satisfaction let me write what i have learnt. Please let me know whether i am absolutely correct or not.

Rotation of an object about a fixed point refers to its motion about the point in such a way that the distance of each of the particles comprising it from that point remains constant. This point is known as the **point of rotation or the central point**. A perpendicular passing through the point of rotation is known as the **axis of rotation**. Each point on the axis of rotation remains stationary.

Axial vectors/pseudovectors are cross products of two vectors.

To describe the rotational motion of an object and the direction of vectors related to it (axial vectors) we have to specify the orientation of the axis of rotation along with sense of rotation since there exist infinitely many axes of rotation in 3 dimensions. In 2 dimensions, however, there exists only one axis of rotation, which is oriented along the Z-axis, i.e., it is perpendicular to the X-Y plane.

Therefore in 2 dimensions, the orientation of the axis of rotation is always the same and is implicitly implied. To specify the sense of rotation we add a - sign or a + sign to denote clockwise and anticlockwise sense of rotation respectively.

In 3 dimensions, the rotation of an object is possible about any of the infinite
number of axes of rotation and for each axis, there is a corresponding clockwise and anticlockwise sense of rotation. Thus in 3 dimensions, to describe the rotational motion of an object and the direction of vectors related to it, it is necessary to specify both the orientation of the axis of rotation as well as the sense of rotation for that particular axis. The best way to do this is to use the Right Hand Rule according to which, if the two vectors of which the axial vector is a cross product are placed tail to tail and the right hand is placed along the direction of the first vector and the fingers are curled in the sense in which the first vector should be rotated in order to reach the second vector through the smaller angle, then the direction in which the extended thumb points is the direction of the axial vector. This direction encodes the sense of rotation along with the orientation of the axis (which is perpendicular to the plane containing the two vectors which are being cross-multiplied).

Thus - and + signs are not required in 3 dimensions since the sense of rotation is encoded in the direction of axial vectors along with the orientation of the axis.

 

[A.T.]

In 3 dimensions, the rotation of an object is possible about any of the infinite number of axes of rotation and for each axis, there is a corresponding clockwise and anticlockwise sense of rotation.

Again, the concepts clockwise and anticlockwise do not make much sense here. For any 3D axis there a two possible rotation directions, period.

 

[vanhees71]

...and usually one chooses the orientation according to the right-hand rule.

 

[zwierz]

https://en.wikipedia.org/wiki/Tensor_density

 

[PeroK]

Thanks to all! Just for the sake of satisfaction let me write what i have learnt. Please let me know whether i am absolutely correct or not ...

Thus in 3 dimensions, to describe the rotational motion of an object and the direction of vectors related to it, it is necessary to specify both the orientation of the axis of rotation as well as the sense of rotation for that particular axis.

You need only specify the axis of rotation. If, say, the axis of rotation is the z-axis and the rotation is $\theta$ that means $\theta$ "anticlockwise" about the z-axis. More precisely, the axis of rotation is specified by the positive z-axis - or by unit vector $(0, 0, 1)$.

For a "clockwise" rotation about the z-axis, the axis of rotation is the negative z-axis: the unit vector $(0, 0, -1)$. Or, this is equivalent to a rotation of $-\theta$ about the positive z-axis.

In either case, the rotation is entirely specified by the vector $(0, 0, \theta)$, where $\theta$ can be positive or negative, or simply in the range $0 \le \theta < 2\pi$.

In this way, the "orientation" of the axis of rotation is encapsulated simply by the vector describing the rotation.

The way I remember rotations is that if you look down the axis (i.e. with the axis pointing at you), then a positive rotation is anti-clockwise.

The other thing I remember is that $\vec{i} \times \vec{j} = \vec{k}$. Everything to do with cross products follows from that.