# Show that v=ωr using dot product or vector products

• Jan 26th 2013, 01:21 AM
happymatthematics
Show that v=ωr using dot product or vector products
Show that v=ωr using dot product or vector products.
anyone can teach me how to do the question below?

Attachment 26705
• Jan 26th 2013, 06:53 AM
Hartlw
Re: Show that v=ωr using dot product or vector products
v = wR = rsin(theta)w. v = wXr, (right hand rule)
• Jan 26th 2013, 03:57 PM
happymatthematics
Re: Show that v=ωr using dot product or vector products
Thank you Hartlw. but I still don't understand.
sorry to ask you more questions.
do R has a physical meaning? I don't know why v = wR.
also, I only learnt ||aXb||=||a||||b||sin(thetre), but didn't learn what aXb is.
• Jan 26th 2013, 06:30 PM
HallsofIvy
Re: Show that v=ωr using dot product or vector products
$\displaystyle v= \omega r$ doesn't make any sense until you define what those symbols mean! That formula could have many meaning but I suspect it is the relation of the "angular speed" of an object moving in a circle to its "linear speed". If so, yes, "r" has a "physical meaning". It is the radius of the circle. If an object is moving in a circle of radius r, with angular speed, $\displaystyle \omega$, its linear speed (speed in m/s around the circumference) is $\displaystyle v= \omega r$. That follows pretty much from the definition of radian measure- the length of a section of a circle of radius r having central angle $\displaystyle \theta$ is $\displaystyle \theta r$. If r is constant, then the speed with which that length changes is the speed with which $\displaystyle \theta$ changes (which is what "angular speed" means) times the constant radius.

I have no idea why you would learn "||aXb||=||a||||b||sin(theta)" without learning what aXb meant! That's like learning that the "absolute value of codswallop" is 4 without learning what "coswallop" meant!

The cross product of two vectors, aXb, is a vector perpendicular both vectors a and b, having length ||a||||b||sin(theta) where ||a|| is the length of vector a and ||b|| is the lengthof vector v.

You seem to be learning a number of formulas without learning what they mean. That's not a good idea.
• Jan 27th 2013, 02:24 PM
happymatthematics
Re: Show that v=ωr using dot product or vector products
Thank you, HallsofIvy.^^
yes, you're right. Most of the time, I recites things without understanding them.
do you mind I ask you one more question?
v = ωR = ωrsin(theta) = ωXr, where R is the radius of the circle
thus, v=ωXr
It seems that in ωrsin(theta), ω and r are scalars,
but in ωXr, ω and r are vectors, why?