I have a simple function of the form $$ f(\hat{\boldsymbol{x}},\hat{\boldsymbol{y}}) = \hat{\boldsymbol{x}}\cdot\boldsymbol{a} + \hat{\boldsymbol{y}}\cdot\boldsymbol{b} + \hat{\boldsymbol{x}}\cdot\hat{\boldsymbol{y}}c $$ of two unit vectors and I want to find the extrema of this function, but am somehow stuck, by the fact that these are unit vectors.

Is there an elegant vector-based solution? Or do I have to express each unit vector by spherical polar coordinates ...? Perhaps one can construct a function $\tilde{f}(\boldsymbol{x},\boldsymbol{y})$ whose extrema occur for $\boldsymbol{x}$ and $\boldsymbol{y}$ being unity and coincide with those of $f(\hat{\boldsymbol{x}},\hat{\boldsymbol{y}})$.

  • $\begingroup$ You could start by noting that $\bf{\hat{x}\cdot a}$ is the projection of $\bf{a}$ on $\bf{\hat{x}}$, i.e. is simply the $x$-component of $\bf{a}$... $\endgroup$
    – Demosthene
    Mar 14, 2015 at 13:42
  • $\begingroup$ @Demosthene $\hat{\boldsymbol{x}}$ is one of the function arguments and not a constant unit vector. $\endgroup$
    – Walter
    Mar 14, 2015 at 14:18
  • $\begingroup$ Still, $\bf{\hat{x}\cdot{a}}$ is the projection of $\bf{a}$ on $\bf{\hat{x}}$, therefore the component of $\bf{a}$ in the $\bf{\hat{x}}$-direction, whatever that direction actually is. $\endgroup$
    – Demosthene
    Mar 14, 2015 at 14:22
  • $\begingroup$ @Demosthene yes, of course. so what? $\endgroup$
    – Walter
    Mar 14, 2015 at 15:36


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