Hello, Frostking!
There is a plane that just took off and is climbing northwest through still air
at an airspeed of 200 km/hr, and rising at a rate of 300 m/min.
Resolve its velocity vector into components.
The answer is: .$\displaystyle  140.8 i + 140.8 j + 18 k$
Look at the side view of the plane climbing. Code:
B *
 *
 * 200
18  *
 *
 *
C *            * A
x
The hypotenuse $\displaystyle AB = 200,\;BC = 18.$
Hence: .$\displaystyle x\:=\:\sqrt{200^218^2} \:=\:\sqrt{39,\!676} \quad\Rightarrow\quad x \:\approx\: 199.19$
Now look down at the ground. Code:

C *     + E
: * 
: * x 
: *45°
: 45° * 
  +     *   
D A

We see that: .$\displaystyle AD \:=\:AE \:=\:\frac{x}{\sqrt{2}} \:\approx\:140.8$