Question

Cross Product. Write a function to calculate the cross product of two vectors \(\mathbf{V}_{1}\) and \(\mathbf{V}_{2}\) \(\mathbf{V}_{1} \times \mathbf{V}_{2}=\left(V_{y 1} V_{x 2}-V_{v 2} V_{21}\right) \mathbf{i}+\left(V_{21} V_{x 2}-V_{22} V_{x 1}\right) \mathbf{j}+\left(V_{41} V_{v 2}-V_{x 2} V_{v 1}\right) \mathbf{k}\) where \(\mathbf{V}_{1}=V_{x 1} \mathbf{i}+V_{x 1} \mathbf{j}+V_{z 1} \mathbf{k}\) and \(\mathbf{V}_{2}=V_{x 2} \mathbf{i}+{y_{12}} \mathbf{j}+y_{x 2} \mathbf{k}\). Note that this function will return a real array as its result. Use the function to calculate the cross product of the two vectors \(V_{1}=[-2,4,0.5]\) and \(V_{2}=\) \([0.5,3,2]\).

Short answer

The cross product of the given vectors \(V_1 = [-2, 4, 0.5]\) and \(V_2 = [0.5, 3, 2]\) is \(\mathbf{V}_1 \times \mathbf{V}_2 = [-8, 1, 4.5]\).

Step-by-step solution

Define the given vectors

Let the two vectors be: \(\mathbf{V}_1 = \begin{bmatrix} -2 \\ 4 \\ 0.5 \end{bmatrix}\) and \(\mathbf{V}_2 = \begin{bmatrix} 0.5 \\ 3 \\ 2 \end{bmatrix}\). **Step 2: Write the function**

Define the function to compute the cross product

Let's define a function called "cross_product" that takes two vectors as input and calculates the cross product using the given formula: def cross_product(V1, V2): i = V1[1] * V2[2] - V1[2] * V2[1] j = V1[2] * V2[0] - V1[0] * V2[2] k = V1[0] * V2[1] - V1[1] * V2[0] result = [i, j, k] return result **Step 3: Compute the cross product for the given vectors**

Calculate the cross product using the function

Now, let's use the function "cross_product" to compute the cross product of the given vectors \(V_1\) and \(V_2\): V1 = [-2, 4, 0.5] V2 = [0.5, 3, 2] cross_product_result = cross_product(V1, V2) The cross product result will be: \(V_1 \times V_2 = [-8, 1, 4.5]\) The cross product is a real array, as required in the problem.

Key concepts

MATLAB Programming

MATLAB, short for Matrix Laboratory, is a high-performance language specifically designed for technical computing. It integrates computation, visualization, and programming in an easy-to-use environment where problems and solutions are expressed in familiar mathematical notation.

When it comes to MATLAB programming, writing functions is essential. Functions allow you to encapsulate and reuse code, making your programs more modular and easier to debug. In the context of computing the cross product of two vectors, a function enhances clarity and maintenance. Understanding how to manipulate arrays and matrices is core to MATLAB, given that even individual vectors are treated as one-dimensional matrices.

For students learning MATLAB, mastering programming constructs such as loops, conditionals, and functions is crucial. However, MATLAB's rich set of built-in functions can often streamline what would otherwise be complex tasks in other programming languages.

Vector Calculations

Vector calculations are fundamental to various disciplines including physics, engineering, and computer graphics. In the provided textbook exercise, we consider the cross product, an operation that takes two vectors and returns another vector that is perpendicular to both.

When performing vector calculations, we must pay attention to components, direction, and magnitude. The cross product is particularly useful because it provides information about the orientation of the two original vectors relative to each other. The resultant vector's magnitude is also a measure of the parallelogram spanned by the two original vectors. It's important for students to visualize these concepts to fully grasp the significance of the cross product.

The arithmetic to find the cross product involves taking the product of the components of the vectors in a specific manner, which can seem complex but is made manageable with practice and understanding the underlying geometric principles.

Programming Functions

In programming, functions are defined blocks of code that perform a specific task and can be reused throughout a program. In our MATLAB example, the function `cross_product` encapsulates the computation of the cross product of two vectors.

The use of functions not only makes the code more readable but also reduces the likelihood of errors. By focusing all the logic to compute the cross product within a single function, you assure consistency in its execution. This is a clear demonstration of the importance of programming functions in algorithmic problem solving.

Learning how to write and use functions efficiently is a key aspect of programming education. The benefits include code reusability, easier debugging, and the ability to divide complex problems into simpler, manageable sub-problems—it's a skill that every student should strive to master.

Mathematical Concepts

Mathematical concepts form the foundation of programming functions in scientific computing. Understanding the theory behind operations like the cross product allows programmers to implement them correctly and efficiently.

The cross product itself is a mathematical concept with applications in various fields, such as torque calculation in physics, finding the normal of a plane in 3D modeling, and determining right-hand rule conformity in vector calculus. It involves a bilinear operation on two vectors, resulting in a third vector that is orthogonal to the plane containing the initial vectors.

Mathematical fluency is hence indispensable for students in fields that involve computational mathematics. Programming languages, like MATLAB, offer a powerful toolkit to apply these concepts, but the underlying mathematical principles remain timeless and universal, requiring deep understanding independently of the computational tools available.