HW1 #4c

You need the magnitude of r and dr/dt squared. For this, you can use the dot product - the magnitude of a vector squared is the vector dotted into itself.

|\vec{r}|^2 = \vec{r}\cdot\vec{r}=\left(\vec{a}\,\cos{\omega t} + \vec{b}\,\sin{\omega t}\right)\cdot \left(\vec{a}\,\cos{\omega t} + \vec{b}\,\sin{\omega t}\right) \\
|\vec{r}|^2 = |\vec{a}|^2\,\cos^2{\omega t} + 2\vec{a}\cdot\vec{b} \,\sin{\omega t}\cos{\omega t} + |\vec{b}|^2\,\sin^2{\omega t}

You have terms with "a dot b" in them. Just leave them be. Find dr/dt, repeat ... and those "cross terms' will drop out anyway. The rest should be easy.