Suppose a solid sphere and a hoop are rolled up an incline plane with the same initial velocity. Which object will travel furthest up the plane

The solid sphere

Explanation:

The solid sphere will travel farther up the plane between it and a hoop. This is because for any mass that is given and any radius, the hoop does require more energy than the sphere, in most or even all cases, in order to overcome its rotational inertia i.e Moment of Inertia of hoop > Moment of Inertia of the sphere).

When you look at the formulas, then

mr² > 2/5mr²

Therefore initial kinetic energy given to the shape is used more times for the rotation (spinning) by the hoop than for translation (moving in a straight line up the incline)

no. dot product   =0 when vecs at right angles

multiple allele inheritance is traits controlled by more than 2 alleles (ex: pigeons having 3 alleles of one gene govern their feather color).