Problem calculating rotation matrix around arbitrary axis I'm trying to calculate the rotating matrix around the Z axis in a counter clockwise direction.
In-order to do that I've decided on a vector A [1 0 0] and I wish to rotate it around the z axis so it'll end up as At [0 1 0] (90 degrees rotation around the z axis)
Let's consider our matrix R. Now, what I've done is set At coefficients to the product of the matrix R by the vector A as follows:
At.x = A.x * R00 + A.y * R10 + A.z * R20
At.y = A.x * R01 + A.y * R11 + A.z * R21
At.z = A.x * R02 + A.y * R12 + A.z * R22

Now, replacing the A and At coefficients we get:
0 = 1 * R00 + 0 * R10 + 0 * R20
1 = 1 * R01 + 0 * R11 + 0 * R21
0 = 1 * R02 + 0 * R12 + 0 * R22

Next, to make these equations right I've deducted the following:
R00 must be 0
R01 must be 1
R02 must be 0
others R coefficients are irrelevant to me

Creating the matrix so far:
0     1     0
0     0     0
0     0     0

Since we rotated around the z axis by 90 degrees we can say the following about the earlier deduction of the matrix coefficients:
R00 = cos(θ)
R01 = sin(θ)
R02 = cos(θ)

And finally, applying this to the matrix:
cos(θ)     sin(θ)     cos(θ)
  0          0          0
  0          0          0

After building my matrix, I've tested it here:
Matrix Calculation Online Website
With the following same vector A [1 0 0] and matrix R I ended up with
And got the desired Result of At [0 1 0]
So, summing it all the process I've done, my question is why for every resource I looked for the answer the rotating around the z axis matrix should look like this: Z axis rotation matrix
What have I done wrong / missed calculating to get the same matrix ?
 A: It seems like you've designed a set of equations specifically for the case of $\theta=90^\circ$.  If you want to derive the more general rotation matrix about axis $z$, then you have to allow for any rotation $\theta$.  
Consider rotating some vector like your vector $\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ or slightly more generally $\begin{bmatrix} r \\ 0 \\ 0 \end{bmatrix}$ around the $z$ axis by an arbitrary angle $\theta$.
 (https://en.wikipedia.org/wiki/Rotation_matrix#/media/File:Counterclockwise_rotation.png)
The position of the head of the new vector can be calculated using trigonometry.  It's really the same mathematics as polar coordinates where here you're rotating a radius of $r$.  
So the new $x$ and $y$ coordinates of the vector head position can be found with
$$
x = r \cos(\theta)
$$
$$
y = r \sin(\theta)
$$
Writing this with matrices gives
$$
\begin{bmatrix} x \\ y \end{bmatrix} = 
\begin{bmatrix}
\cos(\theta) & 0 \\
0 & \sin(\theta) 
\end{bmatrix}
\begin{bmatrix}
r \\ 0
\end{bmatrix}
$$
This gives a slightly more general formula but requires the vector start parallel to the $x$ axis.  We can generalize further by considering rotating any vector $\begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}$ in 3D about the $z$ axis.  Using the same trig we can deduce that the new $x$ position of the vector head is
$$
x = v_x \cos(\theta) - v_y\sin(\theta) 
$$
the new $y$ position is 
$$
y = v_x \sin(\theta) + v_y\cos(\theta) 
$$
and since we're rotating around $z$ the $v_z$ value does not change 
$$
z = v_z
$$
which writing with matrices gives you the rotation matrix you'll recognize
$$
\begin{bmatrix}
x \\ y \\ z 
\end{bmatrix} = 
\begin{bmatrix}
\cos(\theta) &  -\sin(\theta) & 0 \\
\sin(\theta) & \cos(\theta) & 0 \\ 
0 & 0 & 1 \\ 
\end{bmatrix}
\begin{bmatrix}
v_x \\ v_y \\ v_z
\end{bmatrix}
$$
