When are the variables allocated and initialized?

please take a look at this code :

template <typename T>
class matrix
{
    private:
        std::vector<T> vec_;

public:
    matrix(size_t rows , size_t cols) : rows_(rows) , cols_(cols) , vec_(rows_ * cols_)
    {

    }       
    size_t rows_;
    size_t cols_;   

};

This is a way to declare a class. I am just wondering where the std::vector is being allocated and where it is being initialized ?

What happens when I declare a variable ? Is the space for it allocated in the stack before the constructor is called or is allocated and initialized in the contructor?

What is the difference between declaring a vector with size 10

{ 
    private:
        std::vector<T> vec_(10);

and calling the constructor of vec_ with the size 10 in the constructor ?

public: 
     matrix() : vec_(10)

I wanted to understand how objects are allocated and initialized in C++.

Also I can create a constructor without calling the constructor of std::vector

public: 
     matrix() {}

What is happening here? Since I am not calling the constructor for the vector does the compiler call it own its own ? Can the vector object utilized ? Or is it called automatically because I declared std::vector to be a variable in my class ?

Also does initializing as std::vector vec(10) , have the same effect as calling resize/reserve. Which is it closer to to ?