Error in linking to friend functions - c++

Question

I have a class 'Vector3' which is compiled successfully. It contains both non-friend and friend functions, for example, to overload * and << operators when Vector3 is the second operand. The problem is I can't link to any of the friend functions, be it operator overloaded or not. So I can confirm that the error is not specific to operator overloading. The g++ command used for linking is as follows (please also see Makefile at the end),
g++ -Wall -W -I./ -g -o main.out main.o Vector3.o
which gave the following errors,
main.cpp:7: undefined reference to `operator*(double, Vector3 const&)'
main.cpp:9: undefined reference to `mag(Vector3 const&)'
main.cpp:10: undefined reference to `operator<<(std::basic_ostream<char, std::char_traits<char> >&, Vector3 const&)'
Below is the relevant code in my source files. I follow the practice of making separate .hpp and .cpp for every class.
/* file Vector3.hpp */
struct Vector3 {
...
Vector3 operator*(const double k) const;
friend Vector3 operator*(const double k, const Vector3 &vec);
double magnitude() const;
friend double mag(const Vector3 &vec);
friend std::ostream& operator<<(std::ostream& output, const Vector3 &vec);
...
}
/* file Vector3.cpp */
Vector3 operator*(const double k, const Vector3 &vec) {
...
}
inline double mag(const Vector3 &vec) {
...
}
std::ostream& operator<<(std::ostream& output, const Vector3 &vec) {
...
}
/* file main.cpp */
#include "Vector3.hpp"
int main() {
Vector3 M(1, 1, 1);
M = M * 2.0; // own operator* links successfully
M = 10.0 * M; // friend operator* doesn't link
double m = M.magnitude(); // own function magnitude() links successfully
double n = mag(M); // friend function mag() doesn't link
std::cout << M; // friend operator<< doesn't link
}
Finally, this is my Makefile.
CXX = g++
CXXFLAGS = -Wall -W $(INCPATH) -g
INCPATH = -I./
OBJS = main.o Vector3.o
main.out: $(OBJS)
$(CXX) $(CXXFLAGS) -o $# $(OBJS) $(LIBPATH)
main.o: main.cpp
Vector3.o: Vector3.cpp
clean:
rm -f $(OBJS) main.out
The strangest thing is that if I include the Vector3.cpp file as well in main.cpp and then remove Vector3.o from OBJS in Makefile, the program links successfully. I cannot make sense of this. Please help me!!

Answer 1

The definition of friend operator* uses fp_type while the friend declaration uses double as the first parameter. This will only work as intended if fp_type is a typedef-name for double. Are you sure fp_type actually stands for double? I can't see it from the code you posted.
The problem with mag is rather obvious: you defined it as inline in .cpp file. Inline function definitions have to be visible everywhere they are used, meaning that normally they should be placed in the header file.