Blog Entry © Tuesday, April 14, 2026, by James Pate Williams, Jr. Exercises and Supplementary Problems from Introduction to Vector Analysis Fourth Edition © 1979 by Harry F. Davis and Arthur David Snider Selected Exercises from Chapter 1 Page 48

// VectorAnalysis.cpp © Tuesday, April 14, 2026
// by James Pate Williams, Jr.
// Reference: Introduction to Vector Analysis Fourth Edition
// © 1979 by Harry F. Davis and Arthur David Snider 

#include <iostream>
#include <vector>

static double InnerProduct(
	std::vector<double> A,
	std::vector<double> B,
	int n)
{
	double sum = 0.0;

	for (int i = 0; i < n; i++)
		sum += A[i] * B[i];

	return sum;
}

static void VectorProduct(
	std::vector<double> A,
	std::vector<double> B,
	std::vector<double>&C)
{
	C.resize(3);
	C[0] = A[1] * B[2] - A[2] * B[1];
	C[1] = A[0] * B[2] - A[2] * B[0];
	C[2] = A[0] * B[1] - A[1] * B[0];
}

static double TripleProduct(
	std::vector<double> A,
	std::vector<double> B,
	std::vector<double> C)
{
	double sum0 = 0.0, sum1 = 0.0;

	sum0 += A[0] * B[1] * C[2] + A[1] * B[2] * C[0] + A[2] * B[0] * C[1];
	sum1 += A[1] * B[0] * C[2] + A[0] * B[2] * C[1] + A[2] * B[1] * C[0];
	return sum0 - sum1;
}

static void Exercises_Section_1_13_1_Triple_Products()
{
	// triple products (a)
	std::vector<double> A1 = { 2, 0, 0 };
	std::vector<double> B1 = { 0, 3, 0 };
	std::vector<double> C1 = { 0, 0, 5 };
	double tp_1 = TripleProduct(A1, B1, C1);
	std::cout << "Exercise 1 (a)" << '\t';
	std::cout << tp_1 << std::endl;
	// triple products (b)
	std::vector<double> A2 = { 1, 1, 1 };
	std::vector<double> B2 = { 3, 1, 0 };
	std::vector<double> C2 = { 0, -1, 5 };
	std::cout << "Exercise 1 (b)" << '\t';
	double tp_2 = TripleProduct(A2, B2, C2);
	std::cout << tp_2 << std::endl;
	// triple products (c)
	std::vector<double> A3 = { 2, -1, 1 };
	std::vector<double> B3 = { 1, 1, 1 };
	std::vector<double> C3 = { 2, 0, 3 };
	double tp_3 = TripleProduct(A3, B3, C3);
	std::cout << "Exercise 1 (c)" << '\t';
	std::cout << tp_3 << std::endl;
	// triple products (d)
	std::vector<double> A4 = { 0, 0, 1 };
	std::vector<double> B4 = { 1, 0, 0 };
	std::vector<double> C4 = { 0, 1, 0 };
	double tp_4 = TripleProduct(A4, B4, C4);
	std::cout << "Exercise 1 (d)" << '\t';
	std::cout << tp_4 << std::endl;
	// volume of a parallelpipped
	std::vector<double> A5 = { 3, 4, 1 };
	std::vector<double> B5 = { 2, 3, 4 };
	std::vector<double> C5 = { 0, 0, 5 }; 
	double tp_5 = TripleProduct(A5, B5, C5);
	std::cout << "Exercise 2" << '\t';
	std::cout << tp_5 << std::endl;
	// volume of a parallelpipped
	std::vector<double> A6 = { 3, 2, 1 };
	std::vector<double> B6 = { 4, 2, 1 };
	std::vector<double> C6 = { 0, 1, 4 };
	std::vector<double> D6 = { 0, 0, 7 };
	std::vector<double> AB = { -1, 0, 0 };
	std::vector<double> AC = { 3, 1, -3 };
	std::vector<double> AD = { 3, 2, -6 };
	std::cout << "Exercise 3" << '\t';
	double tp_6 = TripleProduct(AB, AC, AD);
	std::cout << tp_6 << std::endl;
	// volume of a tetrahedron
	std::vector<double> AB1 = { 1, 1, 0 };
	std::vector<double> AC1 = { 1, -1, 0 };
	std::vector<double> AD1 = { 0, 0, 2 };
	std::cout << "Exercise 4" << '\t';
	double tp_7 = TripleProduct(AB1, AC1, AD1) / 6.0;
	std::cout << fabs(tp_7) << std::endl;
	std::vector<double> P1 = { 0, 0, 0 };
	std::vector<double> P2 = { 1, 1, 0 };
	std::vector<double> P3 = { 3, 4, 0 };
	std::vector<double> P4 = { 4, 5, 0 };
	std::vector<double> P5 = { 0, 0, 1 };
	std::vector<double> Q1 = { 1, 1, 0 };
	std::vector<double> Q2 = { 2, 3, 0 };
	std::vector<double> Q3 = { 1, 1, 1 };
	std::cout << "Exercise 5" << '\t';
	double tp_8 = TripleProduct(Q1, Q2, Q3);
	std::cout << fabs(tp_8) << std::endl;
	std::vector<double> A10 = { 1, 1, 1 };
	std::vector<double> B10 = { 2, 4, -1 };
	std::vector<double> C10 = { 1, 1, 3 };
	double tp_10 = TripleProduct(A10, B10, C10);
	std::vector<double> D10 = { 0 };
	VectorProduct(A10, B10, D10);
	double magnitude = sqrt(InnerProduct(D10, D10, 3));
	std::cout << "Exercise 10" << '\t';
	std::cout << tp_10 / magnitude << '\t';
	std::cout << 2.0 * sqrt(38.0) / 19.0 << std::endl;
}

int main()
{
	Exercises_Section_1_13_1_Triple_Products();
	return 0;
}