Deriving angular velocity vector algebra?

[SpartanG345]

where x represents cross product

currently if i forget i figure these out using the right hand rule, but how do you get each equation visa versa using vector algebra

i started with w = rxv

how do you derive that v = wxr

i got up to this

w = rxv
w= -(vxr)
rxw = -rx(vxr)
rxw = v(r.r) - r(r.v)
rxw= v -r(r.v)

but can u assume r.v are perpendicular?
is this the right approach? and you even relate the 2 equations this way??

 

[jav]

This is not true in general.

We know that w=rxv mus be orthogonal to both r and v by definition.

If v=wxr, then v would be orthogonal to both w and r. We know v is orthogonal to w by hypothesis, but v is not necessarily orthogonal to r.

Bottom line, you would have to assume orthogonality of r and v to make any kind of assertion like this.

 

[Ateacher]

You were wrong from the start I'm afraid.

For v_t is tangential velocity (theta_hat component)

w=v_t/r
v_t=|v_t|=|rhat x v|
sometimes but not always more useful:
rhat=r/r and
v_t=|(r/r) x v|
in vector form then:
w= (rhat/r) x v=(r/r^2) x v

then
wr= rhat x v

in absolute value this is:
wr=v_t

Of course in absolute value you could have just skipped all the vector stuff,
so presumably it is the vector result that you are interested in.

 

[Ateacher]

BTW it is true that:
wxr=v_t
where v_t=v_t theta_that=rhatxv

Since all three are now orthogonal , proof of that comes from unit vector cross product rules, basically the right hand rule anyway, except it will work for a right or left handed rule since w depends in the first place on which rule you're using. That's why if one side of an equal is an axial vector, the other side also should also , and also why, as in the case here, an axial vector crossed with a vector, is a vector. The result of that cross product doesn't depend on the rule.

 

[CellCoree]

First of all, it is important to note that the equations you have provided are not entirely correct. The correct equations for angular velocity using vector algebra are:

1. w = r x v
2. w = -(v x r)

To derive the second equation from the first, we can use the properties of the cross product and vector algebra. Let's start with the first equation:

w = r x v

To get v on one side, we can cross both sides with r:

r x w = (r x r) x v

Since the cross product of a vector with itself is zero, we can simplify the right side to just v:

r x w = v

Now, we can use the fact that the cross product is anti-commutative, meaning that a x b = -(b x a). So we can rewrite the equation as:

w x r = -(v x r)

And finally, we can rearrange the terms to get the second equation:

w = -(v x r)

This approach is correct, but it's important to note that r and v are not necessarily perpendicular. They are just two vectors in three-dimensional space. The cross product is used to find a vector that is perpendicular to both r and v, and that is why it is used in the equations for angular velocity.

In conclusion, vector algebra can be used to derive the equations for angular velocity, but it's important to use the correct properties of the cross product and understand the relationship between the vectors involved.