Question

Let p= sin t i + cost j + tk, What is |dp/dt|?

Answer 1

$$\vec{p}=(\sin{t})\vec{i}+(\cos{t})\vec{j}+t\vec{k}$$

where $\vec{i}, \vec{j}, \vec{k}$ are unit vectors.

Finding a derivative:

$\frac{d\vec{p}}{dt}=\frac{d(\sin{t})}{dt}\vec{i}+\frac{d(\cos{t})}{dt}\vec{j}+\frac{dt}{dt}\vec{k}=(\cos{t})\vec{i}-(\sin{t})\vec{j}+\vec{k}$

Finding a module for an arbitrary vector

$$\vec{a}=a_x\vec{i}+a_y\vec{j}+a_z\vec{k}$$

is found by the formula:

$$|\vec{a}|=\sqrt{a_x^2+a_y^2+a_z^2}$$

Finding a module:

$|\frac{d\vec{p}}{dt}|=\sqrt{(\cos{t})^2+(-\sin{t})^2+1^2}=\sqrt{2}$