If you want to rotate a vector you should construct what is known as a rotation matrix.
Rotation in 2D
Say you want to rotate a vector or a point by θ, then trigonometry states that the new coordinates are
x' = x cos θ − y sin θ
y' = x sin θ + y cos θ
To demo this, let's take the cardinal axes of X and Y; when we rotate the X-axis 90° counter-clockwise, we should end up with the X-axis transformed into Y-axis. X-axis as a unit-vector is
unit vector along X axis = <1, 0>
x' = 1 cos 90 − 0 sin 90 = 0
y' = 1 sin 90 + 0 cos 90 = 1
New coordinates of the vector = <x', y'> = <0, 1> => Y-axis.
When you understand this, creating a matrix to do this becomes simple. A matrix is just a mathematical tool to perform this in a comfortable, generalized way so that various transformations like rotation, scale and translation (moving) can be combined and performed in a single step, using one common method. From linear algebra, to rotate a point or vector in 2D, the matrix to be built is
|cos θ −sin θ| |x| = |x cos θ − y sin θ| = |x'|
|sin θ cos θ| |y| |x sin θ + y cos θ| |y'|
Rotation in 3D
That works in 2D, while in 3D we need to take in to account the third axis. Rotating a vector around the origin (a point) in 2D simply means rotating it around the Z-axis (a line) in 3D; since we're rotating around Z-axis its coordinates should kept constant i.e. 0° (the rotation happens on the XY plane in 3D). In 3D rotating around the Z-axis would be
|cos θ −sin θ 0| |x| |x cos θ − y sin θ| |x'|
|sin θ cos θ 0| |y| = |x sin θ + y cos θ| = |y'|
| 0 0 1| |z| | z | |z'|
around the Y-axis would be
| cos θ 0 sin θ| |x| | x cos θ + z sin θ| |x'|
| 0 1 0| |y| = | y | = |y'|
|−sin θ 0 cos θ| |z| |−x sin θ + z cos θ| |z'|
around the X-axis would be
|1 0 0| |x| | x | |x'|
|0 cos θ −sin θ| |y| = |y cos θ − z sin θ| = |y'|
|0 sin θ cos θ| |z| |y sin θ + z cos θ| |z'|
Note that the axis around which rotation is done has no sin or cos elements in the matrix. I hope this makes the rotation case clear.
Composition
The aforementioned matrices rotate an object as if the object is at a distance r = √(x² + y²) from the origin; lookup polar coordinates to know why. This rotation will be with respect to the world space origin. Usually we need to rotate an object around its own frame/pivot and not around the world's. Since not all objects are at the world origin, rotating using these matrices will not give the desired result of rotating around the object's own frame. Hence you need to learn about translation too. You'd first translate (move) the object to world origin (so that the object's origin would align with the world's, thereby making r = 0), perform the rotation with one (or more) of these matrices and then translate it back again to its previous location. The order in which the transforms are applied matters.
I urge you to read about linear and affine transformations and their composition to perform multiple transformations in one shot, before playing with transformations in code. Without understanding the basic maths behind it, debugging transformations would be a nightmare. I found this lecture video to be a very good resource. Another resource is this tutorial on transformations that aims to be intuitive and illustrates the ideas with animation.
Note: This method of performing rotations follows the Euler angle rotation system, which is simpler to teach and to grasp. This works perfectly fine for 2D and for simple 3D cases; but when rotation needs to be performed around all three axes at the same time then Euler angles are not sufficient for this due to an inherent deficiency in this system which manifests itself as Gimbal lock. People resort to Quaternions in such situations, which is more advanced than this but doesn't suffer from Gimbal locks when used correctly.
Best Answer
This is a problem that 3D animation software has to deal with all the time, namely when an object's coordinate system is related to another object's coordinate system rather than the global coordinate system. In the general case, there may be arbitrarily many such coordinate systems each adding their own pitch/yaw/roll to each child object.
The solution used by Blender (https://www.blender.org) is based on Quaternions. Wikipedia explains how quaternions work in this case: https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation. That explanation is perhaps too lengthy to be copied here (though it can be if that's the only way to validly "answer" the question).