This might seem like a stupid question, but
why is theta in the follow picture
$60^\circ$, I understand how it is $60^\circ$ through simple trig, through $2 \cos \theta$, you can find $\theta$ to be $60^\circ$. Yet if you do the equation
$$
\frac{|u\cdot v|}{||u||\cdot ||v||} = \cos\theta
$$
while $v = [1,2]$ and $u = [1,0]$, you get $\theta \approx 63^\circ$.
What am I doing wrong?
P.S. This comes from a question I got where:
Bert can swim at a rate of $2$ miles per hour in still water. The current in a river is flowing at a rate of $1$ mile per hour. If Bert wants to swim across the river to a point directly opposite, at what angle to the bank of the river must he swim?
I know the answer is 60*, yet I am confused why it is 60 but not around 63. Thank you.
Best Answer
Your vectors are wrong. In particular, the vector $v$ is not $(1,2)$ but $(1,\sqrt{3})$ so that its magnitude (i.e. the speed of swimming) would be $2$.
With $v=(1,\sqrt{3})$, you have $u\cdot v=1$ and $|u|=1, |v|=2$ so you get $\cos\theta=1/2$ as expected.