Position and Velocity vector

Question

How do you find the velocity vector if the position vector is given by $\langle \sec t, \tan t \rangle$ ? I am familiar with the method of deriving position to get velocity but this is given as a vector itself not an equation.

Answer 1

Differentiate position with respect to time component-wise to get the velocity vector, which is what you asked for: $$\mathbf{\vec{v}} = \left\langle \frac{d}{dt}(\sec t) , \frac{d}{dt}(\tan t) \right\rangle$$

It’s just like if you wrote position in terms of constant unit vectors and then differentiated that:

$$\begin{align} \mathbf{\vec r} &= \mathbf{\hat i} \sec(t) + \mathbf{\hat j} \tan(t) \\ \frac{d\mathbf{\vec r}}{dt} &= \frac{d}{dt}(\mathbf{\hat i} \sec t) + \frac{d}{dt}(\mathbf{\hat j} \tan t) \\ \mathbf{\vec v} &= \mathbf{\hat i}\frac{d}{dt}( \sec t) + \mathbf{\hat j}\frac{d}{dt}( \tan t) \end{align}$$