Answer to Question #101556 in Classical Mechanics for israel

Question

10. If a is a unit vector which is a function of a scalar prove that da/dt, in general a and da/dt are at right angles.

Answer 1

Let's us assume unit vector as $a=cos \ \theta \ \hat { i }\ +sin\ \theta\ \hat{j}$

calculating the $da/dt\ =-(sin\ \theta)\ \hat{i}\ +cos\ \theta\ \hat{j}$

so the vector $a$ and $da/dt$ are perpendicular if their dot product are zero

$a.da/dt=0$

hence vector $a$ and $da/dt$ are perpendicular.