What happens to an object moving near speed of light being accelerated perpendicularly to the direction of travel? Let's say we have an object traveling in space with some velocity $\vec{v}=(v_x,v_y,v_z)=(c, 0, 0)$ and we apply a force to this object perpendicularly to its direction, say $\vec{F}=(0,y,0)$.
What happens to the velocity of this object?
$\lvert v\rvert$ can not exceed $c$, so does $v_x$ decrease in order to compensate for the increase in $v_y$ or does $v_y$ not change at all? (Or does something else happen?)
I have (crudely) drawn an image to illustrate what I am asking for:

The upper part is a photon's path being bent by a black hole, and the second is a space ship traveling near $c$ and then being accelerated perpendicularly to the direction of travel.
 A: At such a great speed, the relativistic mass should be considered, in F = ma. Then, you will realise that the force will cause negligible acceleration. As a result, you will not be able to accelerate it enough (in any direction), to exceed the speed of light. The direction doesn't really matter.
A: The behavior of a a spaceship moving upwards at speed v, with a rocket on its side blasting horizontally, as in the OP's picture, is such that the upwards speed stays constant, while the horizontal speed approaches but never reaches $$\sqrt {c^2-v^2} $$
The behavior of a a spaceship moving upwards at speed v, with a force pushing the spaceship horizontally, as in the OP's picture, is such that the upwards speed approaches but never reaches zero, while the horizontal speed approaches but never reaches c.
If an object moves horizontally as in the OP's upper picture, and a some force, like gravity, pulls said object vertically, then the vertical speed increases approaching c, while the horizontal speed decreases approaching zero.
