Calculate the angular momentum of this particle in rotational motion

[YanZhen]

Homework Statement

The mass of the particle is m,the equation of motion is r=acosωti+bsinωtj.a and b are constant.Calculate the angular momentum of a particle at any time.

Relevant Equations

L=P*r=m*v*r
v=dr/dt

i,j,k arevector
I know L=P*r=m*v*r=m(acosωti+bsinωtj)*(-aωsinωti+bωcosωtj)=mabw((cos^2)ωt+(sin^2)ωt)k=mabωk.
but why m(acosωti+bsinωtj)*(-aωsinωti+bωcosωtj)=mabw((cos^2)ωt+(sin^2)ωt)k.I need some detail.
please help me.

 

[Orodruin]

You should not be using regular multiplication to multiply vectors, you need to use the vector valued cross product to define the angular momentum:
$$
\vec L = \vec p \times \vec r
$$
where
$$
\hat i \times \hat j = \hat k, \quad \hat j \times \hat k = \hat i, \quad \hat k \times \hat i = \hat j
$$
and furthermore you have anti-symmetry so $\vec u \times \vec v = - \vec v \times \vec u$ for any vectors $\vec u$ and $\vec v$.

 

[BvU]

IIRC $$\vec L\equiv \vec r \times\vec p$$

 

[Orodruin]

IIRC $$\vec L\equiv \vec r \times\vec p$$

Too busy arguing about the need to use the cross product to care about getting the right definition I suppose ...

 

[YanZhen]

You should not be using regular multiplication to multiply vectors, you need to use the vector valued cross product to define the angular momentum:
$$
\vec L = \vec p \times \vec r
$$
where
$$
\hat i \times \hat j = \hat k, \quad \hat j \times \hat k = \hat i, \quad \hat k \times \hat i = \hat j
$$
and furthermore you have anti-symmetry so $\vec u \times \vec v = - \vec v \times \vec u$ for any vectors $\vec u$ and $\vec v$.

wow,thanks for your wonderful answer,it solved my question.and it also requires a point of knowlege about vectors.(x1,y1)X(x2,y2)=x1y2-y1x2

 

[Orodruin]

(x1,y1)X(x2,y2)=x1y2-y1x2

Careful. This is not the cross product. It is a component (the third) of the cross product between two vectors that only have i and j components. The cross product as such does not exist in two dimensions (although there are generalisations of it that do - among them the wedge product, which would look something like what you wrote but is generally too advanced for basic introduction to vectors and their application). The general form of the cross product (in three dimensions) would read
$$
(x_1,y_1,z_1) \times (x_2, y_2, z_2) = (y_1 z_2 - y_2 z_1, z_1 x_2 - z_2 x_1, x_1 y_2 - x_2 y_1)
$$