Set your coordinate axis such that positive X is east and positive Y is north (therefore negative X is west and negative Y is south).
Then break them each into components as carteblanche said. When a vector points northeast or northwest, it makes a 45 degree angle with both the X and Y axes, so multiply the resultant (the value of the vector) by cos(45) for x and sin(45) for y.
A = (x component, y component)
A = (23.4*cos(45), 23.4*sin(45))
A = (16.5, 16.5)
B = (-15.3*cos(45), 15.3*sin(45))
Here the x component is negative since it points west.
B = (-10.8, 10.8)
Now the vector operations are easy.
A-B = (16.5 - (-10.8), 16.5 - 10.8)
A-B = (27.3, 5.7)
To find the length, use the pythagorean theorem:
sqrt(27.3^2 + 5.7^2) = 27.9
B - A found similarly
A - 5B = (16.5 - 5(-10.8), 16.5 - 5(10.8))
-1.5A = (-1.5*16.5, -1.5*16.5)