/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 Cross Product. Write a function ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 10

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

Expert verified
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

01

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**
02

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**
03

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攊t'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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Sort with Carry. It is often useful to sort an array are1 into ascending order, while simultancously carrying along a second array arr2. In such a sort, each time an element of array arrl is exchanged with another element of arr1, the corresponding elements of array arr2 are also swapped. When the sort is over, the elements of array arri are in ascending order, whereas the elements of array arr 2 that were associated with particular elements of array arrl are still associated with them. For example, suppose we have the following two arrays. \(\begin{array}{ccc}\text { Element } & \text { arr1 } & \frac{\operatorname{arc} 2}{1 .} \\ 1 . & 6 . & 0 . \\ 2 . & 1 . & 10 . \\\ 3 . & 2 . & 10 .\end{array}\) After sorting array arrl while carrying along array arr2, the contents of the two arrays will be \(\begin{array}{ccc}\text { Element } & \text { arr1 } & \text { are2 } \\ 1 . & 1 . & 0 . \\ 2 . & 2 . & 10 . \\ 3 . & 6 . & 1 .\end{array}\) Write a function to sort one real array into ascending order while carrying along a second one. Test the function with the following two 9 -element arrays. $$ \begin{aligned} &a=[-1, \quad 12, \quad-6, \quad 17, \quad-23,0,5, \quad 1,-1]: \\ &b=[-31,101,36,-17, \quad 0,10,-8,-1,-1]= \end{aligned} $$

Use the Help Browser to look up information about the standard MATLAB function sortrows and compare the performance of sortrows with the sort-with-carry function created in the previous exercise. To do this, create two copies of a \(1000 \times 2\) element array containing random values, and sort column 1 of each array while carrying along column 2 using both functions. Determine the execution times of each sort function using tic and toe. How does the speed of your function compare with the speed of the standard function sorerows?

Gaussian (Normal) Distribution. Function randomo returns a uniformly distributed random variable in the range \([0,1)\), which means that there is an equal probability of any given number in the range occurring on a given call to the function. Another type of random distribution is the Gaussian distribution, in which the random value takes on the classic bellshaped curve shown in Figure 5.9. A Gaussian distribution with an average of \(0.0\) and a standard deviation of \(1.0\) is called a standardized normal distribution, and the probability of any given value occurring in the standardized normal distribution is given by the equation $$ p(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2} $$ It is possible to generate a random variable with a standardized normal distribution starting from a random variable with a uniform distribution in the range \([-1,1)\) as follows. 1\. Select two uniform random variables \(x_{1}\) and \(x_{2}\) from the range \([-1,1)\) such that \(x_{1}^{2}+x_{2}^{2}<1\). To do this, generate two uniform random variables in the range \([-1,1)\), and see if the sum of their squares happens to be less than \(1.0\). If so, use them. If not, try again. 2\. Then each of the values \(y_{1}\) and \(y_{2}\) in the equations that follow will be a normally distributed random variable. $$ \begin{aligned} &y_{1}=\sqrt{\frac{-2 \ln r}{r}} x_{1} \\ &y_{2}=\sqrt{\frac{-2 \ln r}{r} x_{2}} \end{aligned} $$ where $$ r=x_{1}^{2}+x_{2}^{2} $$ a In is the natural logarithm (log to the base e). Write a function that returns a normally distributed random value cach time that it is called. Test your function by getting 1000 random values, calculating the standard deviation, and plotting a histogram of the distribution. How close to \(1.0\) was the standard deviation?

Derivative in the Presence of Noise. We will now explore the effects of input noise on the quality of a numerical derivative. First, generate an input vector containing 100 values of the function \(\sin x\) starting at \(x=0\), using a step size \(\Delta x\) of \(0.05\), just as you did in the previous problem. Next, use function randomo to generate a small amount of random noise with a maximum amplitude of \(\pm 0.02\), and add that random noise to the samples in your input vector (see Figure 5.8). Note that the peak amplitude of the noise is only \(2 \%\) of the peak amplitude of your signal, since the maximum value of \(\sin x\) is 1. Now take the derivative of the function using the derivative function that you developed in the last problem. How close to the theoretical value of the derivative did you come?

Modify the selection sort function developed in this chapter so that it accepts a second optional argument, which may be either ' up ' or "down'. If the argument is 'up', sort the data in ascending order. If the argument is 'dewn', sort the data in descending order. If the argument is missing, the default case is to sort the data in ascending order. (Be sure to handle the case of invalid arguments, and be sure to include the proper help information in your function.)
See all solutions

Recommended explanations on Psychology Textbooks

Memory Studies in Psychology

Read Explanation

Famous Psychologists

Read Explanation

Personality in Psychology

Read Explanation

Addiction

Read Explanation

Sensation and Perception

Read Explanation

Biological Bases of Behavior

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.