I was doing a physics problem and found that I reached an interesting equation:
To cross the river directly from A to B (making a right angle with the velocity of the stream), how must the fisherman direct his/her boat's motion? Find the angle ($\alpha$) between the velocity vector of the stream and the velocity vector of the boat. The magnitude of the velocity of the boat is $v=2.778\mathrm{ms}^{-1}$ and that of the stream is $u=1.3889\mathrm{ms}^{-1}$.
Solution:
We know from the Parallelogram law of addition of vectors:
$$\tan\theta=\frac{Q\sin\alpha}{P+Q\cos\alpha}$$
Similarly,
$$\tan(90^{\circ})=\frac{u\sin\alpha}{v+u\cos\alpha}$$
$$\implies\frac{\sin(90^{\circ})}{\cos(90^{\circ})}=\frac{u\sin\alpha}{v+u\cos\alpha}$$
$$\implies \frac{1}{0}=\frac{u\sin\alpha}{v+u\cos\alpha}$$
$$\implies{v+u\cos\alpha}=0$$
Now, isn't $\;\dfrac10\;$ undefined? How is cross-multiplication valid here? I'm very confused.
Best Answer
It is not valid per se, because as you pointed out, cross multiplication with $\frac{1}{0}$ is to be avoided.
However, a better way to put it, is that a rational expression $$\frac{u\sin\alpha}{v+u\cos\alpha}$$ is equal to $\tan(\pi^r/2)$ which is undefined in the real domain, and the only way a rational function of the form $\frac{f(x)}{g(x)}$ can be undefined is when $g(x) = 0$ (i.e) $$ v + u\cos\alpha = 0 $$
Hope this helps.