Derivative of a unit vector

[mech-eng]

Homework Statement

the first derivative of a vector of constant magnitude is the cross product of the angular velocity
of the vector(i.e , the angular velocity of the moving coordinate system ) and the vector itself.)

Homework Equations

di/dt= w x i , here w is the angular velocity and x is the cross product symbol

The Attempt at a Solution

dİ/dt=di/dΘ * dΘ/dt because i(unit vector in the direction i) is changing only in direction i.e it is
rotating, only it's direction will change.its direction changes with Θ(the angle) and Θ changes with
with the time. so there must be a chain rule. but I can't show that di/dt=wxi

 

[voko]

What is the definition of "the angular velocity of the vector"?

 

[lightgrav]

would be more explicit to write that unit vector using trig functions of the angle theta ... then take derivatives.

 

[mech-eng]

What is the definition of "the angular velocity of the vector"?

unit vector is rotating so angular velocity must be dΘ/dt

 

[vanhees71]

It's much easier to prove that the time derivative of a unit vector is always perpendicular to itself. Then it's clear that the derivative must be of the form $\dot{\vec{i}}=\vec{\omega} \times \vec{i}$, which defines the angular velocity.

 

[voko]

unit vector is rotating so angular velocity must be dΘ/dt

Even assuming that $\theta$ is defined (which it is not), that does not define a vector.

I'd suggest that you follow the advice of vanhees71.

 

[mech-eng]

It's much easier to prove that the time derivative of a unit vector is always perpendicular to itself. Then it's clear that the derivative must be of the form $\dot{\vec{i}}=\vec{\omega} \times \vec{i}$, which defines the angular velocity.

yes , it is much easier to show that, but it can be another orthogonal forms such ixw or ixj or
jxi these three are also perpendicular to it.

 

[mech-eng]

Even assuming that $\theta$ is defined (which it is not), that does not define a vector.

I'd suggest that you follow the advice of vanhees71.

but there are two coordinates in the plane which unit vector i lies , Θ and L (polar coordinates),
since it is a unit vector its magnitude is constant and thus L isn't a coordinate. this shows that
there must be an angle which describes the rotation of unit vector i so we must assume that Θ is defined. Mean while I have found a geometrical approach but I am much interested in analytical or algebraic methods.

 

[voko]

yes , it is much easier to show that, but it can be another orthogonal forms such ixw or ixj or
jxi these three are also perpendicular to it.

That is not true. If you have two orthogonal vectors, there is only one (in 3D) vector, up to a scaling factor, that is orthogonal to both of them. Because the scaling factor can be negative, there are only two unit vectors with that property; they have opposite directions. For angular velocity, a particular direction is chosen, that is part of its definition.

 

[mech-eng]

That is not true. If you have two orthogonal vectors, there is only one (in 3D) vector, up to a scaling factor, that is orthogonal to both of them. Because the scaling factor can be negative, there are only two unit vectors with that property; they have opposite directions. For angular velocity, a particular direction is chosen, that is part of its definition.

What is a scaling factor for vectors or of vectors? I can't find any basic info about it on the web.

 

[mech-eng]

What is the definition of "the angular velocity of the vector"?

This is a true question related with my question. How does w appear in the equation as i'=i x w
and does it appear from other relations between w, direction angle Θ, and the time?

,

 

[voko]

If two vectors $\vec a$ and $\vec b$ are such that $\vec a = c \vec b$, where $c$ is a number, then we say that the vectors are equal up to a scaling factor, and the scaling factor is $c$ Geometrically this means the vectors are collinear.

If both vectors are unit vectors, then $c = \pm 1$.

 

[voko]

This is a true question related with my question.

You need to be able to answer this clearly and unambiguously. Otherwise nothing will ever make sense.

 

[D H]

voko is spot on here.

mech-eng, you have been a bit sloppy in this thread. What do you mean by $\vec{\omega}$? By $\hat \imath$? When you ask about $\frac {d \hat \imath}{dt}$, with respect to what is your $\hat \imath$ expressed?

We can't guess at the underlying nature of these questions, but they're just guesses. It would be much better if you filled in those blanks for us. If you can't, is there a textbook whose notation is confusing to you? If that's the case, a reference to the text and the pages within it would be helpful.

 

[haael]

I get the general idea mech-eng wants to grasp.

Suppose we have a vector dependent on time, with constant (unit) length but with variable angle. The vector is wandering happily over the surface of the unit sphere.

The question: what will be the time derivative of the variable vector? What mathematical object will it be?

The problem will be easier if we rewrite it slightly. Suppose we have the variable vector:
$v(t)$
Now, let's take the value of the vector in the time 0.
$v_0 = v(0)$
Now let's consider rotation matrices. There is always a rotation matrix that transforms the original vector into the current vector
$v(t) = R(t) v_0$

Now let's consider the time derivative of this expression. Since $v_0$ is constant, we have:
$\frac{∂v(t)}{∂t} = \frac{∂R(t)}{∂t} v_0$

To calculate the derivative of the rotation matrix, let's consider the following:
First, rotation matrices form a group. Multiplication of two rotation matrices yields another rotation matrix.
$R(t + Δt) = (1 + S(t) Δt) R(t)$ (where 1 is the unit matrix and $S(t)$ is some other matrix)
$R(t + Δt) = R(t) + S(t) R(t) Δt$
$R(t + Δt) - R(t) = S(t) R(t) Δt$
$\frac{R(t + Δt) - R(t)}{Δt} = S(t) R(t)$
By taking the limit $Δt → 0$, we get the derivative:
$\frac{∂R(t)}{∂t} = S(t) R(t)$

Now what is the S(t) matrix?
It must satisfy the first equation:
$R(t + Δt) = (1 + S(t) Δt) R(t)$

R(t + Δt) and R(t) are rotation matrices. If this equation has to hold, then S(t) must be a skew-symmetric matrix.

Let's go back to the original problem:
$v(t) = R(t) v_0$
We can write now (in the limit Δt → 0):
$v(t + Δt) = R(t + Δt) v_0$
$v(t + Δt) = (1 + S(t) Δt) R(t) v_0$
$v(t + Δt) = (1 + S(t) Δt) v(t)$
$v(t + Δt) = v(t) + S(t) v(t) Δt$
$v(t + Δt) - v(t) = S(t) v(t) Δt$
$\frac{v(t + Δt) - v(t)}{Δt} = S(t) v(t)$
$\frac{∂v(t)}{∂t} = S(t) v(t)$

So, the derivative of the vector wandering over the surface of the sphere will be equal to the vector multiplied by the skew-symmetric matrix that can be computed from the relevant rotation matrix.

Now, in 3 dimensions we have a special situation. The skew-symmetric matrix has exactly 3 independent components that can be rearranged into something that resembles a vector:
$S(t) = \begin{matrix} 0 & a & -b \\ -a & 0 & c \\ b & -c & 0 \end{matrix}$
$s(t) = [a, b, c]$
This is called a pseudovector or an axial vector and means what is known under the name "angular velocity".

The operation of the pseudovector on the original vector is given by the cross product:
$S(t) v(t) = s(t) \times v(t)$

So, in the Δt → 0 limit:
$v(t + Δt) = v(t) + (s(t) \times v(t)) Δt$

This is what you mean by saying "vector v(t) has the angular velocity s(t)".

 

[mech-eng]

You need to be able to answer this clearly and unambiguously. Otherwise nothing will ever make sense.

it should, of course, be changing of direction with respect to time and using trig function(which lightgrav said) we can define a unit vector. Let it be in xy plane and in positive x direction. we should specify this by defining angle θ whose derivative should be size of angular velocity with respect to time. For that unit vector being in position x direction we should choose θ as 0 and then unit vector m=cos θi+sinθj, in terms of fixed coordinate vectors, then we can take its derivative which is,dm/dt=-sinΘi*dΘ/dt+cosΘj*dΘ/dt.Then by factoring out dm/dt=dΘ/dt(-sinΘi+cosΘj) and inserting Θ=0 and dΘ/dt=w it becomes dm/dt=wj which is equal of w x i. Is it a true approach?

 

[voko]

it should...

I cannot follow that. I do not know what you have as a given and what you are supposed to derive. I do not even know what definition of "angular velocity" you have, or even if you have one to begin with. It's been a week but we have not made much progress.

Please give us the exact definition of "angular velocity of a vector" that you are supposed to use.

 

[voko]

Now let's consider rotation matrices.

There is no need to jump back and forth between vectors and matrices. It is a good exercise after you have learned both, but it only confuses matters while you are under way.

It is sufficient that for a unit vector $\vec u \cdot \vec u = 1$. Then you have immediately $\vec u \cdot \dot {\vec u} = 0$ which means $\dot {\vec u}$ is orthogonal to $\vec u$. This in turn means that we can define another unit vector orthogonal to these two vectors (using either the right-hand or the left-hand rule). This is the direction of the angular velocity. The magnitude of the angular velocity is simply $\dot u / u$. The rules of cross product then imply that $\dot {\vec u} = \vec \omega \times \vec u$.

 

[haael]

The rules of cross product then imply that

Yes, but where do the rules of cross product come from?

There is no need to jump back and forth between vectors and matrices. It is a good exercise after you have learned both, but it only confuses matters while you are under way.

In my opinion, one must first learn vectors and translations, then matrices, then the rotation group, then skew-symmetric matrices and then the cross-product approach to angular motion.
This was the way I understood this. Your path may be different.

I think it's wrong to teach about cross products without any justification where they come from.

 

[D H]

Then you have immediately $\vec u \cdot \dot {\vec u} = 0$ which means $\dot {\vec u}$ is orthogonal to $\vec u$. This in turn means that we can define another unit vector orthogonal to these two vectors ...

Picky detail: $\vec u \cdot \dot {\vec u}$ means that $\dot {\vec u}$ is either zero or orthogonal to $\vec u$. If $\dot{\vec u}$ is zero (i.e., $\vec u$ is constant, period, not just constant magnitude), you can't use $\vec u$, $\dot {\vec u}$, and $\vec u \times \dot {\vec u}$ to span three dimensional space.

 

[mech-eng]

There is no need to jump back and forth between vectors and matrices. It is a good exercise after you have learned both, but it only confuses matters while you are under way.

It is sufficient that for a unit vector $\vec u \cdot \vec u = 1$. Then you have immediately $\vec u \cdot \dot {\vec u} = 0$ which means $\dot {\vec u}$ is orthogonal to $\vec u$. This in turn means that we can define another unit vector orthogonal to these two vectors (using either the right-hand or the left-hand rule). This is the direction of the angular velocity. The magnitude of the angular velocity is simply $\dot u / u$. The rules of cross product then imply that $\dot {\vec u} = \vec \omega \times \vec u$.

But here , how did you define angular velocity? You just showed u and derivative of u is orthogonal to each other and then defined a third vector orhogonal to them admitting that direction of that third vector and angular velocity is the same.Then you specified magnitude of angular velocity but while you are specifying it you used u'=w x u. the problem was arriving at u'=w x u but you used u'=w x u at the beginning and checked it. I will find the book again and then try to make concepts more clear.

 

[voko]

Yes, but where do the rules of cross product come from?

There are many ways to introduce the cross product. I do not see a problem in defining the cross product as a vector orthogonal to two given vectors and with the magnitude equal to the area of the parallelogram formed by those two vectors.

One feature of this approach is that the compact and simple vector algebra can be mastered by a student relatively quickly in its entirety; matrix algebra, rotations, etc are a lot more material to cover.

 

[voko]

But here , how did you define angular velocity? You just showed u and derivative of u is orthogonal to each other and then defined a third vector orhogonal to them admitting that direction of that third vector and angular velocity is the same.Then you specified magnitude of angular velocity but while you are specifying it you used u'=w x u. the problem was arriving at u'=w x u but you used u'=w x u at the beginning and checked it.

No, I did not use $\dot {\vec u} = \omega \times \vec u$ to specify the magnitude of $\omega$. I said it was $\dot u / u$. In fact I should have said it was $\dot u$ because we are dealing with a unit vector.

Anyway, angular velocity is indeed defined via $\vec u$ and $\dot {\vec u}$. I do not think it can be defined without $\dot {\vec u}$, but you can try - that is why we keep asking for the definition.

 

[mech-eng]

No, I did not use $\dot {\vec u} = \omega \times \vec u$ to specify the magnitude of $\omega$. I said it was $\dot u / u$. In fact I should have said it was $\dot u$ because we are dealing with a unit vector.

Anyway, angular velocity is indeed defined via $\vec u$ and $\dot {\vec u}$. I do not think it can be defined without $\dot {\vec u}$, but you can try - that is why we keep asking for the definition.

I have just looked up at R.C Hibbeler's Engineering Mechanics -Dynamics book(not the book when I started the topic). Writer uses a geometric approach. In the book i and j unit vectors are drawn orthogonal to each other then turning these i and j vectors by dΘ , new vectors i' and j' are constructed.In the limit when Δt goes dt di is tangent to the path defined by tip of i thus in the direction of j. And dj is in the direction of -i. this is what is written in that dynamic book. I think this is a geometric approach and writer doesn't explain why di is tangent to the path defined
by tip of i while i is rotated by dΘ. But, I think, writing a unit vector, say vector m, as trigonometric components in i and j directions and then taking derivative is a better approach.

 

[voko]

Do you have a specific question?

 

[ehild]

Homework Statement

the first derivative of a vector of constant magnitude is the cross product of the angular velocity
of the vector(i.e , the angular velocity of the moving coordinate system ) and the vector itself.)

Homework Equations

di/dt= w x i , here w is the angular velocity and x is the cross product symbol

The Attempt at a Solution

dİ/dt=di/dΘ * dΘ/dt because i(unit vector in the direction i) is changing only in direction i.e it is
rotating, only it's direction will change.its direction changes with Θ(the angle) and Θ changes with
with the time. so there must be a chain rule. but I can't show that di/dt=wxi

I think this can be useful for you. http://www.chem.ox.ac.uk/teaching/Physics%20for%20CHemists/Rotation/vectors.html

ehild