A 2D vector (u, v) has its base at the origin in the cartesian coordinate system, i.e. thex-yaxes,anditstip(orarrow)atx=uandy=v. Thus,a 2Dvectorcanbe represented simply as an (x,y) point in the cartesian coordinate system. Write a program that reads in a pair of 2D vectors and a scalar value and then applies the following vector operations: addition, subtraction, scalar multiplication, and perpendicularity. The main routine of the program is already provided in vector2D_template.cpp (which you copied into vector2D.cpp). Your task is to add eight functions read_vector(), vector_length(), write_vector(), vector_add(), vector_subtract(), scalar_mult(), normalize(), and perpendicular() so that the program produces the desired results. 1. Write your code incrementally. I HIGHLY SUGGEST that you write each function one at a time as described in the following steps. After you have written a solution to 1 a particular function, you should compile, run, and test your code before moving to the next step. 2. DO NOT MODIFY ANY OF THE CODE in procedure main. 3. DO NOT change the global constant called EPSILON defined at the top. 4. You will use the global constant EPSILON whenever you need to determine the equality of two values of type double (Please study the lecture notes before you continue). For example, if a and b are values of type double you can use the following condition to determine that they are equal: abs(a b) < EPSILON, where abs() is the absolute value function. 5. A function prototype should be written for each function and placed BEFORE the main procedure. 6. All functions should be written AFTER the main procedure. 7. Each function should have a comment explaining what it does. 8. Each function parameter should have a comment explaining the parameter. 9. Write a function read_vector() which inputs from the user a 2D vector by reading its x- and y-coordinates. Define the function so that it does not return a value and has three input parameters: a string to hold the prompt displayed to the user. Define the string as a constant parameter and use pass by reference. See the lecture notes on how to do this. a number x of type double representing the first coordinate of the 2D vector. Define this parameter as pass by reference. a number y of type double representing the second coordinate of the 2D vector. Define this parameter as pass by reference. 10. Write a function vector_length() which returns the length of a 2D vector. The length of vector (x, y) is x2 + y2. Input to the function are the coordinates x and y, which are passed by value. The function returns the length as type double. 11. Write a function write_vector() which outputs a 2D vector and its length. Call the function vector_length() to compute the length. Define the function so that it does not return a value and has three input parameters: a string to hold the message to be displayed before the coordinates of the vector. Define the string as a constant parameter and use pass by reference. a number x of type double representing the first coordinate of the 2D vector. Define this parameter as pass by value. 2 a number y of type double representing the second coordinate of the 2D vector. Define this parameter as pass by value. 12. Write a function vector_add() which adds two 2D vectors and results in a new 2D vector. To add two vectors (x1, y1) and (x2, y2) you will simply add the corresponding coordinate values, i.e. (x1, y1) + (x2, y2) = (x3, y3) where x3 = x1 + x2 and y3 = y1 + y2. Define the function so that it does not return a value and has six input parameters: a number x1 of type double representing the first coordinate of the first 2D vector. Define this parameter as pass by value. a number y1 of type double representing the second coordinate of the first 2D vector. Define this parameter as pass by value. a number x2 of type double representing the first coordinate of the second 2D vector. Define this parameter as pass by value. a number y2 of type double representing the second coordinate of the second 2D vector. Define this parameter as pass by value. a number x3 of type double representing the first coordinate of the resultant 2D vector. Define this parameter as pass by reference. a number y3 of type double representing the second coordinate of the resultant 2D vector. Define this parameter as pass by reference. 13. Write a function vector_subtract() which subracts two 2D vectors and results in a new 2D vector. To subract use the formula (x1, y1) (x2, y2) = (x3, y3) where x3 = x1 x2 and y3 = y1 y2. Like function vector_add, this function also returns no value and uses the same input parameters (see instructions above). 14. Write a function scalar_mult() which applies scalar multiplication to a two 2D vector and results in a new 2D vector. Scalar multiplication is defined as s(x1, y1) = (x2, y2), where s is a scalar value, (x2, y2) is the resultant vector, and x2 = s x1 and y2 = s y1. Define the function so that it does not return a value and has five input parameters: a number x1 of type double representing the first coordinate of the first 2D vector. Define this parameter as pass by value. a number y1 of type double representing the second coordinate of the first 2D vector. Define this parameter as pass by value. a number s of type double representing the scalar multiplier. Define this param- eter as pass by value. a number x2 of type double representing the first coordinate of the resultant 2D vector. Define this parameter as pass by reference. 3 a number y2 of type double representing the second coordinate of the resultant 2D vector. Define this parameter as pass by reference. 15. Write a function normalize() which normalizes vector (x, y) by dividing by its length. Normalization results in a new vector that has the same direction but has a length of 1, called a unit vector. Use the following formula to compute the resuling vector: x=x x2+y2 y=y x2+y2 Call the function vector_length() to compute the length. If the length = 0, then usetheformulaabove. Iflength=0,thensetx=0andy=0toavoidadivisionby zero. Since the length is type double you will need to determine its equality with zero using EPSILON as shown in step 4 above. Let a be the length and b be zero. Define the function so that it does not return a value and has two parameters that are both passed by reference: a number x of type double representing the first coordinate of the 2D vector; a number y of type double representing the second coordinate of the 2D vector. 16. Write a function perpendicular() which determines whether two 2D vectors are perpendicular to each other. In order to make this determination, we will compute two perpendicular vectors from a given vector (x,y). The first perpendicular vector, (px1,py1), is defined as px1 = y and py1 = x. The second perpendicular vector, (px2, py2), is defined as px2 = px1 and py2 = py1. For example if (x, y) = (1, 2) then (px1,py1) = (2,1) and (px2,py2) = (2,1). Thus, (px1,py1) and (px2,py2) are perpendicular vectors to (x, y). You must implement the following algorithm to determine if two vectors, (x1, y1) and (x2, y2), are perpendicular: (a) Normalize (x1,y1) and (x2,y2); Lets call the new vectors v1 = (vx1,vy1) and v2 = (vx2, vy2), respectively (use better variable names). Call the normalize() function to compute these. Note that v1 and v2 are unit vectors. (b) Compute two perpendicular vectors to v1 (see instructions above). Lets call the two new vectors p1 and p2. (c) Check whether v2 is the same as either p1 or p2. If so, then output Vectors are PERPENDICULAR. to the screen. Otherwise, output Vectors are NOT PERPENDICULAR. See next step for more details. (d) Let p1 = (px1, py1), and p2 = (px2, py2). 4 i. To determine if v2 is the same vector as p1, check that both vx2 = px1 and vy2 = py1. Use EPSILON as described in step 4 above to determine each equality. ii. To determine if v2 is the same vector as p2, check that both vx2 = px2 and vy2 = py2. Use EPSILON as described in step 4 above to determine each equality. Define the function so that it does not return a value and has four parameters that are all passed by value: a number x1 of type double representing the x coordinate of the first 2D vector. a number y1 of type double representing the y coordinate of the first 2D vector. a number x2 of type double representing the x coordinate of the second 2D vector; a number y2 of type double representing the y coordinate of the second 2D vector.

This is what I have so far.

#include #include #include using namespace std; const double EPSILON(1e-12); // function prototypes // ENTER FUNCTION PROTOTYPE FOR read_vector HERE. // ENTER FUNCTION PROTOTYPE FOR vector_length HERE. // ENTER FUNCTION PROTOTYPE FOR write_vector HERE. // ENTER FUNCTION PROTOTYPE FOR vector_add HERE. // ENTER FUNCTION PROTOTYPE FOR vector_subtract HERE. // ENTER FUNCTION PROTOTYPE FOR scalar_mult HERE. // ENTER FUNCTION PROTOTYPE FOR normalize HERE. // ENTER FUNCTION PROTOTYPE FOR perpendicular HERE. // *** DO NOT CHANGE ANY CODE IN THE MAIN FUNCTION. int main() { double u1, v1; // coordinates of first vector double u2, v2; // coordinates of second vector double u3, v3; double scalar; read_vector("Enter first vector (2 floats): ", u1, v1); read_vector("Enter second vector (2 floats): ", u2, v2); cout << "Enter scalar multiplier: "; cin >> scalar; cout << endl; write_vector("First vector: ", u1, v1); write_vector("Second vector: ", u2, v2); cout << endl; vector_add(u1, v1, u2, v2, u3, v3); write_vector("Vector add: ", u3, v3); vector_subtract(u1, v1, u2, v2, u3, v3); write_vector("Vector subtract: ", u3, v3); scalar_mult(u1, v1, scalar, u3, v3); write_vector("Scalar multiplier: ", u3, v3); cout << endl; write_vector("First vector: ", u1, v1); write_vector("Second vector: ", u2, v2); perpendicular(u1, v1, u2, v2); return(0); } // DEFINE FUNCTION read_vector HERE. // DEFINE FUNCTION vector_length HERE. // DEFINE FUNCTION write_vector HERE. // DEFINE FUNCTION vector_add HERE. // DEFINE FUNCTION vector_subtract HERE. // DEFINE FUNCTION scalar_mult HERE. // DEFINE FUNCTION normalize HERE. // DEFINE FUNCTION perpendicular HERE.