opengl - Basic Car Movement, Converting 2D Vectors to 3D problems

I'm trying to follow this guys method for steering a basic car. Link.

It's the first example I can sort of get my head around and I'm trying to convert it to a working 3D version.

To get some scope, I've rendered a cube in OpenGL which represents a car and I'm using GLM for matrix manipulation.

Even though I'm trying to get a 3D example working, I'm only using 2D controls; the X axis for left/right and Z axis for forwards/backwards.

For clarity, I have shown all the vectors as vec.x, vec.y, vec.z. note that not much happens with the vec.y as for the time being the car stays level on the X Z plane. (like top-down games but in 3D environment).

I'm getting some strange responses on screen and I can't see why. I can only assume I've misunderstood where some of the code belongs.

These are my variables etc:

glm::vec3 carLocation(0,0,0);   //location of car
float carHeading    = 0.0f;     //direction car facing
float carSpeed      = 0.5f;     //speed of car
float steeringAngle = 0.0f;     //angle of front wheel
float wheelBase;                //length of car
glm::vec3 frontWheel;           //vec 3 representing front wheel
glm::vec3 backWheel;            //vec 3 representing back wheel

glm::vec3 sincos_Calc;          //vec 3 representing sin/cos calculation

This is my function to render the car and manipulate it:

void renderMovingCube(){
glUseProgram(movingCubeShader.handle());
GLuint matrixLoc4MovingCube = glGetUniformLocation(movingCubeShader.handle(), "ProjectionMatrix");  
glUniformMatrix4fv(matrixLoc4MovingCube, 1, GL_FALSE, &ProjectionMatrix[0][0]);

    glm::mat4 viewMatrixMovingCube;
    viewMatrixMovingCube = glm::lookAt(camOrigin, camLookingAt, camNormalXYZ);


        //set values
        wheelBase = 12;                     //car dim X;
        sincos_Calc.x = cos(carHeading);    //adjust X coord
        sincos_Calc.y = 0;                  //car stays on Y plane
        sincos_Calc.z = sin(carHeading);    //adjust Z coord

        //step 1
        frontWheel.x = (carLocation.x + (wheelBase/2)) * sincos_Calc.x;
        frontWheel.y = 0;

        frontWheel.z = (carLocation.z + (wheelBase/2)) * sincos_Calc.z;
        backWheel.x  = (carLocation.x - (wheelBase/2)) * sincos_Calc.x;
        backWheel.y  = 0;
        backWheel.z  = (carLocation.z - (wheelBase/2)) * sincos_Calc.z;

    //location of car
    ModelViewMatrix = glm::translate(viewMatrixMovingCube,carLocation);

    ModelViewMatrix = glm::translate(ModelViewMatrix,  glm::vec3(0,-48,0));
    ModelViewMatrix = glm::rotate(ModelViewMatrix, steeringAngle, glm::vec3(0,1,0));//manually turn


    glUniformMatrix4fv(glGetUniformLocation(movingCubeShader.handle(), "ModelViewMatrix"), 1, GL_FALSE, &ModelViewMatrix[0][0]); //pass matrix to shader
movingCube.render();        //car
glUseProgram(0);            //turn off the current shader
}

This is my key input:

void keyboard(){
if (FORWARD) //W key
{

    //step 2
    backWheel.x += carSpeed*sincos_Calc.x;
    backWheel.y += 0;
    backWheel.z += carSpeed*sincos_Calc.z;
    frontWheel.x += carSpeed*(cos(carHeading+steeringAngle));
    frontWheel.y += 0;
    frontWheel.z += carSpeed*(sin(carHeading+steeringAngle));

    //step 3
    carLocation.x = (frontWheel.x+backWheel.x)/2;

    carLocation.y = 0;
    carLocation.z = (frontWheel.z+backWheel.z)/2;
    carHeading = atan2((frontWheel.z-backWheel.z), (frontWheel.x-backWheel.x));

    cout<< glm::to_string(carLocation) << endl;
    BACKWARD = false;  
}
if (ROT_LEFT){ //A key
    steeringAngle +=0.5f;
    ROT_LEFT = false;  

}
if (ROT_RIGHT){ //D key
    steeringAngle -=0.5f;
    ROT_RIGHT = false; 
}
}

I've been trying for days to get this to work and done so much googling and SO'ing to avoid posting this but I'm still have problems with it.

My aim is to have my cube moving forwards/backwards and steering/turing like a car, described in the link above.

Am I even remotely heading in the right direct? I'm not quite sure why Step 2 overwrites step 1, I feel this is why I'm going wrong...

Have I converted the vecs from vec2's to vec3's correctly? and Are my "steps" in the right locations? Thanks.