how do you figure a physics problem with vectors when only given one vector?

Well, you're really given three vectors.

One points up and to the right, length 50 miles. I assume it also tells you the angle that it makes with the horizontal (let's call that "θ" ("theta")).

Another vector represents the first leg of your detour. This vector goes straight to the right. Its tip is vertically aligned with the tip of the diagonal vector. You should be able to see that the length of this vector is (50mi)(cosθ)

The final vector represents the 2nd leg of your detour. It starts at the tip of the "right-pointing" vector, an stretches up to the tip of the diagonal vector. You should be able to see that the length of this vector is: (50mi)(sinθ).

So, add up the lengths of the two horizontal and vertical vectors. You'll have to know the value of "θ" to do this.

Use the Pythagorean Theorem:

a² + b² = c²

a² + b² = 2500.

However, you need more information, such as the angle theta that forms the triangle.

If theta is 45 deg = pi/4 rad,

a = b = 2500/sqrt(2) = 1768 mi each; 3536 mi for the full trip.