0
$\begingroup$

I would like to know how differentiate this equation it should be fairly simple but its been a long night and I cant seem to figure it out.

$r =\frac{f}{R . \hat z} R$

with respect to time.

  • f is some constant (focal length)
  • R is a position vector
  • $R . \hat z $ is inner product

Any help will be appreciated

  • 0
    $\hat z $ is actually just a unit vector in a specific direction (z-axis), R is a position vector2012-10-24

1 Answers 1

0

Assuming the direction of $\hat z$ does not change with time, $ \begin{align} \dot r &= \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{fR}{\langle R, \hat z\rangle}\right)\\ & = \frac{f}{\langle R, \hat z\rangle^2}\left[\langle R, \hat z\rangle \dot R- R\frac{\mathrm{d}}{\mathrm{d}t}\left(\langle R, \hat z\rangle\right)\right]\\ & = \frac{f}{\langle R, \hat z\rangle^2}\left[\langle R, \hat z\rangle \dot R- \langle \dot R, \hat z\rangle R\right] \end{align} $