Question

A two-dimensional vector \(\mathbf{v}\) running from the origin to a point can be represented as a pair consisting of an \(x\)-coordinate and a \(y\)-coordinate. Implement a data abstraction for vectors by giving a constructor make-vect and corresponding selectors xcor-vect and ycor-vect. In terms of your selectors and constructor, implement procedures add-vect, sub-vect, and scale-vect that perform the operations vector addition, vector subtraction, and multiplying a vector by a scalar: $$ \begin{aligned} \left(x_{1}, y_{1}\right)+\left(x_{2}, y_{2}\right) &=\left(x_{1}+x_{2}, y_{1}+y_{2}\right) \\ \left(x_{1}, y_{1}\right)-\left(x_{2}, y_{2}\right) &=\left(x_{1}-x_{2}, y_{1}-y_{2}\right) \\ s \cdot(x, y) &=(s x, s y) \end{aligned} $$

Short answer

Use abstraction to create vectors with `make_vect`, and perform operations using `add_vect`, `sub_vect`, and `scale_vect`.

Step-by-step solution

Define the Constructor

To create the vector abstraction, first define the constructor `make_vect` which takes two arguments, `x` and `y`, representing the coordinates of the vector. This constructor will return a tuple representing the vector: ```python def make_vect(x, y): return (x, y) ```

Define the Selectors

Implement the selectors `xcor_vect` and `ycor_vect` to retrieve the x-coordinate and y-coordinate from the vector representation respectively: ```python def xcor_vect(vect): return vect[0] def ycor_vect(vect): return vect[1] ```

Implement Vector Addition

Define `add_vect` to perform vector addition. This function will take two vectors and return a new vector representing their sum: ```python def add_vect(v1, v2): return make_vect(xcor_vect(v1) + xcor_vect(v2), ycor_vect(v1) + ycor_vect(v2)) ```

Implement Vector Subtraction

Define `sub_vect` to perform vector subtraction. This function will take two vectors and return a new vector representing their difference: ```python def sub_vect(v1, v2): return make_vect(xcor_vect(v1) - xcor_vect(v2), ycor_vect(v1) - ycor_vect(v2)) ```

Implement Vector Scaling

Lastly, implement `scale_vect` to multiply a vector by a scalar. This function will take a scalar and a vector, returning a new scaled vector: ```python def scale_vect(scalar, vect): return make_vect(scalar * xcor_vect(vect), scalar * ycor_vect(vect)) ```

Key concepts

vector mathematics

In mathematics, vectors are crucial for representing objects with both magnitude and direction. If you think about a vector geometrically, it looks like an arrow pointing from one place to another in 2D or 3D space. These are often represented as a pair or a triplet of numbers, such as \( (x, y) \) in two-dimensional vector space.

Vectors can represent various physical quantities like force, velocity, and displacement. This makes them fundamental in physics and engineering. They allow us to perform operations like addition, subtraction, and scaling, which can interpret or transform spatial data effectively.

• Addition: Combine two vectors by adding their respective components.
• Subtraction: Find a vector that represents the difference between two vectors’ respective components.
• Scaling: Multiply a vector by a scalar, altering its magnitude while preserving its direction.

These operations help transform and analyze spatial problems, facilitating tasks ranging from navigation to computer graphics and more.

constructors and selectors

Data abstraction in programming provides a way to handle complex data, making operations more straightforward and coherent. Constructors and selectors are part of this abstraction. They enable us to build and manipulate structured data without exposing internal details.

With respect to vectors, a constructor is used to create vector objects. The `make_vect` function, for example, combines two numbers, representing the x and y coordinates, into a structured vector.

• **Constructor:** A special function that creates an instance of a data structure. For vectors, `make_vect` combines two integers into a vector.

Meanwhile, selectors retrieve information from a data structure. In vectors, selectors like `xcor_vect` and `ycor_vect` provide access to the x-coordinate and y-coordinate, respectively, of the vector.

• **Selectors:** Functions designed to access the components of a data structure. `xcor_vect` and `ycor_vect` are examples that fetch coordinates from a vector.

By using constructors and selectors, data structures become easier to manage, automate, and scale.

vector operations

Vector operations involve manipulating vectors to achieve new results. These operations, such as addition, subtraction, and scaling, are not just mathematic tools, they have practical applications in many fields.

• Vector Addition: This operation combines two vectors resulting in a new vector. By adding corresponding x and y coordinates, a resultant vector is obtained. Mathematically, this is expressed as \( (x_1 + x_2, y_1 + y_2) \).
• Vector Subtraction: Similar to addition, but here you subtract the corresponding coordinates. This finds the difference vector between two vectors and is represented mathematically as \( (x_1 - x_2, y_1 - y_2) \).
• Scale Vector: Multiplying each component of the vector by a scalar alters the vector's magnitude. This is expressed as \( (s \cdot x, s \cdot y) \).

These operations are pivotal in graphical transformations, physics simulations, and in solving complex engineering problems. Understanding them accelerates one’s ability to model and solve real-world issues effectively.