Gram-Shmidt orthogonalization for R3 basis problem

Gram-Shmidt.cpp

#include <iostream>
#include <cmath>
using namespace std;

struct Vector {
	Vector(double x, double y, double z) {
		this->x = x;
		this->y = y;
		this->z = z;
	}

	Vector() : x(0), y(0), z(0) {}

	double x, y, z;

	double length() {
		return sqrt(pow(x, 2) + pow(y, 2) + pow(z, 2));
	}

};


// operator overloading for Vectors

Vector operator * (double alpha, Vector v) {
    return Vector (alpha * v.x, alpha * v.y, alpha * v.z);
}

Vector operator * (Vector v, double alpha) {
	return alpha * v;
}

double operator * (Vector v1, Vector v2) {
	return (v1.x * v2.x + v1.y * v2.y + v1.z * v2.z);
}

Vector operator / (Vector v, double alpha) {
	return Vector (v.x / alpha,  v.y / alpha, v.z / alpha);
}

Vector operator - (Vector v1, Vector v2) {
    return Vector (v1.x - v2.x, v1.y - v2.y, v1.z - v2.z);
}

Vector operator + (Vector v1, Vector v2) {
	return Vector (v1.x + v2.x, v1.y + v2.y, v1.z + v2.z);
}

ostream & operator << (ostream & os, Vector v) {
	os << "[" << v.x << ", " << v.y << ", " << v.z << "]";
	return os;
}


// extra functions for Vectors

Vector createVector() {
	double x, y, z;

	cout << "Define your vectors coordinates: ";
	cin >> x >> y >> z;

	return Vector(x, y, z);
}

/*Vector projection(Vector v1, Vector v2) {

}*/

bool isOrthogonal(Vector v1, Vector v2) {
	return abs(v1 * v2) < 0.0000005;
}

int main()
{
	// non-orthonormal basis vectors
	Vector f1 = createVector();
	Vector f2 = createVector();
	Vector f3 = createVector();

	//orthogonalization
	Vector e1(f1);
    Vector e2 = (f2 - (e1 * (e1 * f2))/(e1 * e1));
	Vector e3 = ( f3 - ( ((f3 * e1) * e1)/(e1 * e1) ) - ( ( (f3 * e2) * e2 )/(e2 * e2) ) );

	//checking orthogonalization
	cout << isOrthogonal(e1, e2) << endl;
	cout << isOrthogonal(e2, e3) << endl;
	cout << isOrthogonal(e1, e3) << endl;

	// normalization
	e1 = e1/e1.length();
	e2 = e2/e2.length();
	e3 = e3/e3.length();

	//checking if normalization worked
	cout << e1.length() << " " << e2.length() << " " << e3.length() << endl;

	//printing out the orthonormal vectors
	cout << e1 << " " << e2 << " " << e3 << endl;


	return 0;
}