How to get a vector from its length and angle

Question

How can I get the vector when I know its length and angle?

E.g. length = 3 and angle = 40°.

Answer 1

If by "angle of the vector" you mean the angle between the vector and the positive side of the $\;x$-axis and counterclockwise, which is the usual agreement, then

$$x=3\cos40^\circ\;,\;\;y=3\sin40^\circ\;,\;\;\text{and the vector is then}\;\;\binom xy$$

Answer 2

In polar coordinates, you already do!

Otherwise draw a picture with horizontal leg of length $x$ and vertical leg of length $y$, take the sine and cosine of the angle and solve for $x$ and $y$.