Find the components along the `x,y,z` axes of the angular momentum `overset rarr(L)` of a particle, whose position vector is `overset rarr(r )` with components `x,y,z` and momentum is `overset rarr(p)` with components `p_(x), p_(y) and p_(z)`. Show that if the particle moves only in the `x-y` plane, the angular momentum has only a z-component.

Explanation

As we know that angular momentum `vecL, vecr and vecp` are related by the relation
`vecL = vecr xx vec p`
In three dimensions `vecL, vecr and vecP` are written as
`L_(x) hati+ L_(y) hatj+L_(g)hatk=(xhati+yhatj+zhatk) xx(p_(x) hati+p_(y) hatj+p_(z) hatk)`
or `L_(x) hati+ L_(y) hatj+ L_(g) hatk=|{:(hati, hatj, hatk),(x,y,z),(p_(x),p_(y),p_(z)):}|`
`=hati(yp_(z)-zp_(y))+hatj(zp_(x)-xp_(z))+hatk(xp_(y)-yp_(x))`
Comparing the coefficient of ij and k on both sides, we get `L_(x)=yp_(z)-zp_(y), L_(y)=zp_(x)-xp_(v)L_(e)=xp_(y)-yp_(x) " "...(i)`
f the particle is moving in x - y plane, then z = 0 and `p_(z)=0`
`:. vecLj= hatk(xp_(y)-yp_(x))`
As `hatk` vector corresponds to the Z-direction, so we find that the angular momentum `vecL` has only the z component.