Show that if a particle moves with constant speed, then the
velocity and acceleration vectors are orthogonal.
If $\displaystyle v(t),\quad a(t) = v'(t),\quad s(t) = \left\| {v(t)} \right\|$ are the velocity, acceleration, and speed functions, then recall that $\displaystyle s'(t) = \frac{{v(t) \cdot v'(t)}}{{\left\| {v(t)} \right\| }} = \frac{{v(t) \cdot a(t)}}{{\left\| {v(t)} \right\| }}$.
But because the speed is constant, it must be the case that $\displaystyle s'(t) = 0$.
That in turn implies that $\displaystyle {v(t) \cdot a(t)}=0$.