PROBLEM:
An airplane flies north with a speed of 500mph.
Then a wind comes from the West with a speed of 75mph.
How fast is the plane traveling now, and how far off course is it (in degrees)?
SOLUTION:
Plane vector:
Magnitude: 500
Direction: <0,1> (Up)
Wind vector:
Magnitude: 75
Direction: <1,0> (Right)

Resultant Plane Vector:
Now we have a right triangle with legs X-axis (length 1) and Y-axis (length 500). Pythagorean Theorem gives hypotenuse length of ~505.6, which is the resultant magnitude.

To get the angle, we use a right-triangle trigonometric ratio of our choice (sin, cos, tan, etc.). I choose sine, which is opposite/hypotenuse. So if ‘x’ is the angle,
sin x = 500 / 505.6
sin x = 0.9889363732
inverse_sine 0.9889363732 = x
x = 81.46924225

So the plane is blown about 81.5 degrees.