Effective use of C++ iomanip library

Only std::setw() is temporary. The other two calls, setiosflags, and setprecision have a permanent effect.

So, you could change your code to :

std::ostream& operator<<(std::ostream &output, const Vector &v){
  output<<"["
    <<std::setiosflags(std::ios::right | std::ios::scientific)
    <<std::setw(23)
    <<std::setprecision(16)
    <<v._x<<", "
    <<std::setw(23)
    <<v._y<<", "
    <<std::setw(23)
    <<v._z<<"]";
  return output;
} 

But now you've borked the flags and precision for the next guy. Try this instead:

std::ostream& operator<<(std::ostream &output, const Vector &v){
  std::ios_base::fmtflags f = output.flags(std::ios::right | std::ios::scientific);
  std::streamsize p = output.precision(16);
  output<<"["
    <<std::setw(23)
    <<v._x<<", "
    <<std::setw(23)
    <<v._y<<", "
    <<std::setw(23)
    <<v._z<<"]";
  output.flags(f);
  output.precision(p);
  return output;
} 

Finally, if you absolutely have to get rid of the duplication of the constant 23, you could do something like this (but I wouldn't recommend it):

struct width {
  int w;
  width(int w) : w(w) {}
  friend std::ostream& operator<<(std::ostream&os, const width& w) {
    return os << std::setw(width.w);
  }
};


std::ostream& operator<<(std::ostream &output, const Vector &v){
  std::ios_base::fmtflags f = output.flags(std::ios::right | std::ios::scientific);
  std::streamsize p = output.precision(16);
  width w(23);
  output<<"["
    <<w
    <<v._x<<", "
    <<w
    <<v._y<<", "
    <<w
    <<v._z<<"]";
  output.flags(f);
  output.precision(p);
  return output;
} 

See also this other question, where they decided that you can't make width permanent.

In C++20 you'll be able to do:

std::ostream& operator<<(std::ostream& output, const Vector& v){
  const int width = 23, precision = 16;
  return output << std::format(
      "[{0:{3}.{4}e}, {1:{3}.{4}e}, {2:{3}.{4}e}]",
      v._x, v._y, v._z, width, precision);
} 

Unlike I/O manipulators, std::format will not change ostream's formatting state saving your from the problem mentioned by Bo Persson:

Your real problem is what happens to the next output after this one...

It will also work I/O manipulators work correctly with Vector.

Until std::format is available you can use the {fmt} library it is based on.

Everything but the setw() actually do that already. They are "sticky".

Your real problem is what happens to the next output after this one...