Rotation of a point around the z axis

I have 3 vectors in three-dimensional space. Name them xaxis, yaxisand zaxis. These vectors are centered around an arbitrary pointsomewhere in three-dimensional space. I'm interested in turning vectors in xaxisand yaxisaround a vector zaxisby a number of degrees θ.

For the following code with values ​​that are arbitrary and immaterial:

double xaxis[3], yaxis[3], zaxis[3], point[3], theta;

How would I rotate xaxisand yaxisabout zaxisin thetadegrees?

Future Note: these attempts do not work. See My Answer for the Correct Solution That Was Found Using BlueRaja-DannyPflughoeft

My matrix based rotation attempt:

double rx[3][3];
double ry[3][3];
double rz[3][3];
double r[3][3];

rx[0][0] = 1;
rx[0][1] = 0;
rx[0][2] = 0;

rx[1][0] = 0;
rx[1][1] = cos(theta);
rx[1][2] = sin(theta);

rx[2][0] = 0;
rx[2][1] = -1.0 * sin(theta);
rx[2][2] = cos(theta);

ry[0][0] = cos(theta);
ry[0][1] = 0;
ry[0][2] = -1.0 * sin(theta);

ry[1][0] = 0;
ry[1][1] = 1;
ry[1][2] = 0;

ry[2][0] = sin(theta);
ry[2][1] = 0;
ry[2][2] = cos(theta);
//No rotation wanted on the zaxis
rz[0][0] = cos(0);
rz[0][1] = sin(0);
rz[0][2] = 0;

rz[1][0] = -1.0 * sin(0);
rz[1][1] = cos(0);
rz[1][2] = 0;

rz[2][0] = 0;
rz[2][1] = 0;
rz[2][2] = 1;

vtkMath::Multiply3x3(rx, ry, r); //Multiplies rx by ry and stores into r
vtkMath::Multiply3x3(r, rz, r); //Multiplies r by rz and stores into r

vtkMath::Multiply3x3(r, xaxis, xaxis);//multiplies a 3x3 by a 3x1
vtkMath::Multiply3x3(r, yaxis, yaxis);//multiplies a 3x3 by a 3x1

This attempt worked only when the plane was in the xy plane:

double x, y;
x = xaxis[0];
y = xaxis[1];
xaxis[0] = x * cos(theta) - y * sin(theta);
xaxis[1] = x * sin(theta) + y * cos(theta);

x = yaxis[0];
y = yaxis[1];
yaxis[0] = x * cos(theta) - y * sin(theta);
yaxis[1] = x * sin(theta) + y * cos(theta);

, BlueRaja-DannyPflughoeft:

double c = cos(theta);
double s = sin(theta);
double C = 1.0 - c;

double Q[3][3];
Q[0][0] = xaxis[0] * xaxis[0] * C + c;
Q[0][1] = xaxis[1] * xaxis[0] * C + xaxis[2] * s;
Q[0][2] = xaxis[2] * xaxis[0] * C - xaxis[1] * s;

Q[1][0] = xaxis[1] * xaxis[0] * C - xaxis[2] * s;
Q[1][1] = xaxis[1] * xaxis[1] * C + c;
Q[1][2] = xaxis[2] * xaxis[1] * C + xaxis[0] * s;

Q[2][0] = xaxis[1] * xaxis[2] * C + xaxis[1] * s;
Q[2][1] = xaxis[2] * xaxis[1] * C - xaxis[0] * s;
Q[2][2] = xaxis[2] * xaxis[2] * C + c;

double x = Q[2][1] - Q[1][2], y = Q[0][2] - Q[2][0], z = Q[1][0] - Q[0][1];
double r = sqrt(x * x + y * y + z * z);

//xaxis[0] /= r;
//xaxis[1] /= r;
//xaxis[2] /= r;

xaxis[0] = x;// ?
xaxis[1] = y;
xaxis[2] = z;