I am trying to learn about velocity vectors but this word problem is confusing me.
A boat is going 20 mph north east, the velocity u of the boat is the durection of the boats motion, and length is 20, the boat's speed. If the positive y axis represents north and x is east the boats direction makes an angle of 45 degrees. You can compute the components of u by using trig
$$u_1 = 20 \cos 45$$
$$u_2 = 20 \sin 45$$
Why? How did this happen? Why sin and why cos? What does this represent? Why two points? What are these two points? It says that these are $R^2$ which I am not sure what that means and my book does not explain. I think t he R means all real numbers and the squared is referencing 2d maybe so x and y but the book doesn't say so I am not so sure. My book mentions none of these things.
Best Answer
$$u_1=20cos(45)$$ is the component of the velocity vector in the x-direction. $$u_2=20sin(45)$$ is the component of the velocity vector in the y-direction.
If you add these two vectors head to tail, then what you get is the velocity vector of the boat (25mph, north-east, i.e. at a 45 degree angle to both the x and y-axes).
Draw the vector out on the x-y plane starting at the origin, now draw the x-component of it and y-component of it. Now you know your hypotenuse is the velocity of the boat in the direction it's travelling in and the angle it makes with x-axis. You use basic trigonometry to obtain the values above (sine = opposite/hypotenuse, cos = adjacent/hypotenuse).
$R^2$ refers to the all points in the xy-plane (I think this is a good enough definition for this purpose). $u_1$ and $u_2$ are vectors.