Will a vector at 45° be larger or smaller than its horizontal and vertical components?
It will be larger
A vector at 45 degrees is the same thing as the hypotenuse of an isosceles right triangle.
So, assume you have a vertical component and a horizontal component each of one unit. By the Pythagorean Theorem, the hypotenuse, which is the magnitude of your 45 degree vector will be
Any vector that is not parallel with one of the independent reference (basis) vectors (often, but not always, taken to lie on the x and y axes in the Euclidean plane, particularly when introducing the idea in a mathematics course) will be larger than its component vectors because of the triangle inequality.
There is a proof in the famous book "Euclid's Elements" for the case of vectors in the two dimensional (Euclidean) plane.
So, taking the positive x and y axes as the respective directions of the horizontal and vertical components:
The vector at 45 degrees is not parallel with either the x or the y axis. Therefore, by the triangle inequality, it is larger than either of its components.