Question

General directions. (1) You may use a computer algebra system (such as Mathematica, Matlab, etc) or a programming language (such as C++, Java, etc) or possibly some other appropriate software of your choice. (2) The words "A point (a vector) is given'' mean that its coordinates (components) are given. The words "A line is given" mean that its equation is given. You can use as an input either equation of the line in some form or its initial point and direction vector. The words "A plane is given'' mean that its general equation is given. You can use as an input either this equation or the vector of coefficients. The words 'Determine ellipsis, find ellipsis" mean that you are asked to create a program(or some kind of software implementation) that takes as input the given data and returns the required equations. (3) Submit a paper that includes: a nice cover sheet, an explanation of the mathematical principles used in your programs, and a printout of sample runs of the programs. The problem. A line l and a plane pi in R^3 are given. (1) Determine whether the line l lies in the plane pi, parallel to this plane, or intersects this plane. (2) If l intersects pi, find the point of intersection.

Not too sure on how to approach this

Answer 1

// vectorCalculus.java

import java.util.*;

public class vectorCalculus {

public static final String divideLine = "\n-----------------------------------------------";

public static void main(String[] args) {

Scanner sc = new Scanner(System.in);

Vector<Double> lineVector = new Vector<Double>();

Vector<Double> planeVector = new Vector<Double>();

System.out.print("Type in the x-coordinate of points in the line: ");

double xCoordinateLine = sc.nextDouble();

System.out.print("Type in the y-coordinate of points in the line: ");

double yCoordinateLine = sc.nextDouble();

System.out.print("Type in the z-coordinate of points in the line: ");

double zCoordinateLine = sc.nextDouble();

System.out.printf("A line is given: (%f, %f, %f)", xCoordinateLine, yCoordinateLine, zCoordinateLine);

System.out.println(divideLine);

System.out.print("Type in the x-coordinate of vector for line: ");

lineVector.add(sc.nextDouble());

System.out.print("Type in the y-coordinate of vector for line: ");

lineVector.add(sc.nextDouble());

System.out.print("Type in the z-coordinate of vector for line: ");

lineVector.add(sc.nextDouble());

System.out.printf("The line vector, d, is given: [%f, %f, %f]", lineVector.get(0), lineVector.get(1),

lineVector.get(2));

System.out.println(divideLine);

System.out.print("Type in the A coefficient value of the plane pi: ");

planeVector.add(sc.nextDouble());

System.out.print("Type in the B coefficient value of the plane pi: ");

planeVector.add(sc.nextDouble());

System.out.print("Type in the C coefficient value of the plane pi: ");

planeVector.add(sc.nextDouble());

System.out.print("Type in the D coefficient value of the plane pi: ");

planeVector.add(sc.nextDouble());

System.out.printf("A plane vector, n, is given: [%f, %f, %f]\n", planeVector.get(0), planeVector.get(1),

planeVector.get(2));

System.out.printf("Plane equation: %fx + %fy + %fz + %f", planeVector.get(0), planeVector.get(1),

planeVector.get(2), planeVector.get(3));

System.out.println(divideLine);

// Parametric equations of line pass thru plane

// formula set for dotProduct, planeEquation, and normOfPlane

double dotProduct = (lineVector.get(0) * planeVector.get(0)) + (lineVector.get(1) * planeVector.get(1))

+ (lineVector.get(2) * planeVector.get(2));

double planeEquation = ((planeVector.get(0) * xCoordinateLine) + (planeVector.get(1) * yCoordinateLine)

+ (planeVector.get(2) * zCoordinateLine) + (planeVector.get(3)));

double normOfPlane = (Math.sqrt(

Math.pow(planeVector.get(0), 2) + Math.pow(planeVector.get(1), 2) + Math.pow(planeVector.get(2), 2)));

if (dotProduct != 0) {

System.out.println("The dot product is not equal to 0. Therefore the line, l, intersects to plane pi.");

System.out.println("The dot product: " + dotProduct);

System.out.println("Creating a line parametric equations for D1.");

System.out.println("For D1,\nx = " + xCoordinateLine + " + " + lineVector.get(0) + "t" + "\ny = "

+ yCoordinateLine + " + " + lineVector.get(1) + "t" + "\nz = " + zCoordinateLine + " + "

+ lineVector.get(2) + "t");

System.out.print("Type in the value t for D1: ");

double valueT1 = sc.nextDouble();

double d1x = xCoordinateLine + lineVector.get(0) * valueT1;

double d1y = yCoordinateLine + lineVector.get(1) * valueT1;

double d1z = zCoordinateLine + lineVector.get(2) * valueT1;

System.out.println("For D1,\nx = " + d1x + "\ny = " + d1y + "\nz = " + d1z);

System.out.println(divideLine);

System.out.println("Creating a line parametric equations for D2.");

System.out.println("For D2,\nx = " + xCoordinateLine + " + " + lineVector.get(0) + "t" + "\ny = "

+ yCoordinateLine + " + " + lineVector.get(1) + "t" + "\nz = " + zCoordinateLine + " + "

+ lineVector.get(2) + "t");

System.out.print("Type in the value t for D2: ");

double valueT2 = sc.nextDouble();

double d2x = xCoordinateLine + lineVector.get(0) * valueT2;

double d2y = yCoordinateLine + lineVector.get(1) * valueT2;

double d2z = zCoordinateLine + lineVector.get(2) * valueT2;

System.out.println("For D2,\nx = " + d2x + "\ny = " + d2y + "\nz = " + d2z);

System.out.println(divideLine);

System.out.println("For P1,\nx = " + d1x + " + " + planeVector.get(0) + "t" + "\ny = " + d1y + " + "

+ planeVector.get(1) + "t" + "\nz = " + d1z + " + " + planeVector.get(2) + "t\n");

System.out.println("For P2,\nx = " + d2x + " + " + planeVector.get(0) + "t" + "\ny = " + d2y + " + "

+ planeVector.get(1) + "t" + "\nz = " + d2z + " + " + planeVector.get(2) + "t");

System.out.println(divideLine);

// Find t

double e1t = (-planeVector.get(3) - planeVector.get(0) * d1x - planeVector.get(1) * d1y

- planeVector.get(2) * d1z)

/ ((planeVector.get(0) * planeVector.get(0)) + (planeVector.get(1) * planeVector.get(1))

+ (planeVector.get(2) * planeVector.get(2)));

System.out.println("t = " + e1t);

double e1x = d1x + planeVector.get(0) * e1t;

double e1y = d1y + planeVector.get(1) * e1t;

double e1z = d1z + planeVector.get(2) * e1t;

System.out.println("E1 = (" + e1x + ", " + e1y + ", " + e1z + ")");

// Find t

double e2t = (-planeVector.get(3) - planeVector.get(0) * d2x - planeVector.get(1) * d2y

- planeVector.get(2) * d2z)

/ (planeVector.get(0) * planeVector.get(0) + planeVector.get(1) * planeVector.get(1)

+ planeVector.get(2) * planeVector.get(2));

System.out.println("t = " + e2t);

double e2x = d2x + planeVector.get(0) * e2t;

double e2y = d2y + planeVector.get(1) * e2t;

double e2z = d2z + planeVector.get(2) * e2t;

System.out.println("E2 = (" + e2x + ", " + e2y + ", " + e2z + ")");

double projectionX = e2x - e1x;

double projectionY = e2y - e1y;

double projectionZ = e2z - e1z;

System.out.println("The projection of line onto plane pi = J: [x, y, z] = [" + e1x + ", " + e1y + ", " + e1z

+ "] + [" + projectionX + ", " + projectionY + ", " + projectionZ + "]t");

}

if (dotProduct == 0 && planeEquation == 0) {

System.out.println(

"The dot product is equal to 0, \nand the initial point of the line, l, satisfies the plane pi equation.");

System.out.println("Therefore the line, l, is in the plane.");

System.out.println("Plane equation: Ax + By + Cz + D = 0");

System.out.println("Plane equation: " + planeVector.get(0) + "x + " + planeVector.get(1) + "y + "

+ planeVector.get(2) + "z + " + planeVector.get(3) + " = 0");

System.out.println("Plane equation: " + planeVector.get(0) + "(" + xCoordinateLine + ") + "

+ planeVector.get(1) + "(" + yCoordinateLine + ") + " + planeVector.get(2) + "(" + zCoordinateLine

+ ") + " + planeVector.get(3) + " = " + planeEquation);

System.out.println(divideLine);

}

if (dotProduct == 0 && planeEquation != 0) {

System.out.println(

"The dot product is equal to 0,\nand the initial point of the line, l, does not satisfy the plane pi equation.");

System.out.println("Therefore the line, l, is parallel to the plane pi.");

System.out.println("Plane equation: Ax + By + Cz + D = 0");

System.out.println("Plane equation: " + planeVector.get(0) + "x + " + planeVector.get(1) + "y + "

+ planeVector.get(2) + "z + " + planeVector.get(3) + " = 0");

System.out.println("Plane equation: " + planeVector.get(0) + "(" + xCoordinateLine + ") + "

+ planeVector.get(1) + "(" + yCoordinateLine + ") + " + planeVector.get(2) + "(" + zCoordinateLine

+ ") + " + planeVector.get(3) + " != 0" + " (" + planeEquation + ")");

System.out.println(divideLine);

System.out.println("Creating a line parametric equations for D1.");

System.out.println("For D1,\nx = " + xCoordinateLine + " + " + lineVector.get(0) + "t" + "\ny = "

+ yCoordinateLine + " + " + lineVector.get(1) + "t" + "\nz = " + zCoordinateLine + " + "

+ lineVector.get(2) + "t");

System.out.print("Type in the value t for D1: ");

double valueT1 = sc.nextDouble();

double d1x = xCoordinateLine + lineVector.get(0) * valueT1;

double d1y = yCoordinateLine + lineVector.get(1) * valueT1;

double d1z = zCoordinateLine + lineVector.get(2) * valueT1;

System.out.println("For D1,\nx = " + d1x + "\ny = " + d1y + "\nz = " + d1z);

System.out.println(divideLine);

System.out.println("Creating a line parametric equations for D2.");

System.out.println("For D2,\nx = " + xCoordinateLine + " + " + lineVector.get(0) + "t" + "\ny = "

+ yCoordinateLine + " + " + lineVector.get(1) + "t" + "\nz = " + zCoordinateLine + " + "

+ lineVector.get(2) + "t");

System.out.print("Type in the value t for D2: ");

double valueT2 = sc.nextDouble();

double d2x = xCoordinateLine + lineVector.get(0) * valueT2;

double d2y = yCoordinateLine + lineVector.get(1) * valueT2;

double d2z = zCoordinateLine + lineVector.get(2) * valueT2;

System.out.println("For D2,\nx = " + d2x + "\ny = " + d2y + "\nz = " + d2z);

System.out.println(divideLine);

System.out.println("For P1,\nx = " + d1x + " + " + planeVector.get(0) + "t" + "\ny = " + d1y + " + "

+ planeVector.get(1) + "t" + "\nz = " + d1z + " + " + planeVector.get(2) + "t\n");

System.out.println("For P2,\nx = " + d2x + " + " + planeVector.get(0) + "t" + "\ny = " + d2y + " + "

+ planeVector.get(1) + "t" + "\nz = " + d2z + " + " + planeVector.get(2) + "t");

System.out.println(divideLine);

// Find t

double e1t = (-planeVector.get(3) - planeVector.get(0) * d1x - planeVector.get(1) * d1y

- planeVector.get(2) * d1z)

/ ((planeVector.get(0) * planeVector.get(0)) + (planeVector.get(1) * planeVector.get(1))

+ (planeVector.get(2) * planeVector.get(2)));

System.out.println("t = " + e1t);

double e1x = d1x + planeVector.get(0) * e1t;

double e1y = d1y + planeVector.get(1) * e1t;

double e1z = d1z + planeVector.get(2) * e1t;

System.out.println("E1 = (" + e1x + ", " + e1y + ", " + e1z + ")");

// Find t

double e2t = (-planeVector.get(3) - planeVector.get(0) * d2x - planeVector.get(1) * d2y

- planeVector.get(2) * d2z)

/ (planeVector.get(0) * planeVector.get(0) + planeVector.get(1) * planeVector.get(1)

+ planeVector.get(2) * planeVector.get(2));

System.out.println("t = " + e2t);

double e2x = d2x + planeVector.get(0) * e2t;

double e2y = d2y + planeVector.get(1) * e2t;

double e2z = d2z + planeVector.get(2) * e2t;

System.out.println("E2 = (" + e2x + ", " + e2y + ", " + e2z + ")");

double projectionX = e2x - e1x;

double projectionY = e2y - e1y;

double projectionZ = e2z - e1z;

System.out.println("The projection of line onto plane pi = J: [x, y, z] = [" + e1x + ", " + e1y + ", " + e1z

+ "] + [" + projectionX + ", " + projectionY + ", " + projectionZ + "]t");

// distance

double distance = Math.abs(((planeVector.get(0) * d1x) + (planeVector.get(1) * d1y)

+ (planeVector.get(2) * d1z) + planeVector.get(3))) / normOfPlane;

System.out.println("The distance between line and pi = " + distance);

}

}

}

================================================================================

sample output

File EditNavigate Search Project Run Window Help y Error Log 의 Tasks E Problems Console 2 <terminated vectorCalculus [Java Application] C:Program Files Java jre7 bin\javaw.exe (04-Jul-2017 7:14:53 AM) Type in the x-koordinate of points in the line: 6 Type in the y-coordinate of points in the line: 8 Type in the z-coordinate of points in the line: 3 A line is given: (6.000000, 8.000000, 3.000000) Type in the x-coordinate of vector for line: 7 Type in the y-coordinate of vector for line: 3 Type in the z-coordinate of vector for line: 5 The line vector, d, is given: [7.000000, 3.000000, 5.000000] Type in the A coefficient value of the plane pi: 7 Type in the B coefficient value of the plane pi: 4 Type in the C coefficient value of the plane pi: 3 Type in the D coefficient value of the plane pi: 8 A plane vector, n, is given: [7.000000, 4.000000, 3.000000 Plane equation: 7.000000x + 4.000000y 3.000000z 8.000000 The dot product is not equal to 0. Therefore the line, 1, intersects to plane pi. The dot product: 76.0 Creating a line parametric equations for D1. For D1, x6.0 7.0t y 8.0 + 3.0t Type in the value t for D1: 2 For D1, x=20.0 у 14.0 Z13.0 Creating a line parametric equations for D2 For D2, x6.0 7.0t y 8.0 + 3.0t Tyne in the value t for D2: 4