${\mathrm{V}}_{\mathrm{b}}={\mathrm{nV}}_{\mathrm{w}}$

To follow the shortest path, the direction of boat has to be at some angle with water flow , the resultant velocity of boat is,

${\mathrm{V}}_{\mathrm{bg}}=\sqrt{{{\mathrm{V}}_{\mathrm{b}}}^2-{{\mathrm{V}}_{\mathrm{w}}}^2}=\sqrt{{\mathrm{n}}^2{{\mathrm{V}}_{\mathrm{w}}}^2-{{\mathrm{V}}_{\mathrm{w}}}^2}={\mathrm{V}}_{\mathrm{w}}\sqrt{{\mathrm{n}}^2-1}$

for a real value of ${\mathrm{V}}_{\mathrm{bg}} , \mathrm{n}>1$

if n > 1 , V_bg > 0 , so, boat can cross the river along shortest path.

Let $\mathrm{\theta }$ is the angle between direction of boat and shortest path.

${\mathrm{V}}_{\perp }={\mathrm{V}}_{\mathrm{b}}\mathrm{cos\theta }={\mathrm{nV}}_{\mathrm{w}}\mathrm{cos\theta } ;$

${\mathrm{V}}_{\parallel }={\mathrm{V}}_{\mathrm{w}}-{\mathrm{V}}_{\mathrm{b}}\mathrm{sin\theta }={\mathrm{V}}_{\mathrm{w}}\left(1-\mathrm{nsin\theta }\right)$

To cross the river, $\mathrm{\theta }>0$.

if n = 1 , ${\mathrm{V}}_{\parallel }\ne 0$ , so boat can not cross the river without drifting.

if $\mathrm{n}\ne 0, {\mathrm{V}}_{\perp }\ne 0$, so, Boat can cross the river whatever is the value of n excluding zero value