If unit vector $\vec c$ makes an angle $\frac{\pi}{3}$ with $\hat i+\hat j$, then minimum and maximum values of $(\hat i×\hat j).\vec c$ respectively are:

$-\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2}$

$\vec c=x\hat i+y\hat j+x\hat k$

$x+y=\sqrt{2}\frac{1}{2}$

$x+y=\frac{1}{\sqrt{2}},y=\frac{1}{\sqrt{2}}-x$

$x^2+y^2+z^2=1,y^2=\frac{1}{2}+x^2-\sqrt{2}x$  $∴z^2=1-x^2-\frac{1}{2}-x^2+\sqrt{2}x=\frac{1}{2}-2x^2+\sqrt{2}x$

$\hat k.\vec c=z$

$x^2+y^2+2xy=\frac{1}{2}$

$z^2_{man}≤\frac{3}{4}=z∈[-\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2}]$