/*! 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 185 For the following exercises, the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 185

For the following exercises, the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are given. a. Find the cross product \(\mathbf{u} \times \mathbf{v}\) of the vectors \(\mathbf{u}\) and v. Express the answer in component form. b. Sketch the vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{u} \times \mathbf{v}\). \(\mathbf{u}=2 \mathbf{i}+3 \mathbf{j}, \quad \mathbf{v}=\mathbf{j}+2 \mathbf{k}\)

Short Answer

Expert verified
Cross product: \((6, -4, 2)\); sketch vectors accordingly.

Step by step solution

01

Identify Vector Components

First, identify the components of each vector. We have the vector \( \mathbf{u} = 2\mathbf{i} + 3\mathbf{j} \) which translates to the components \( \mathbf{u} = (2, 3, 0) \). Similarly, the vector \( \mathbf{v} = \mathbf{j} + 2\mathbf{k} \) has components \( \mathbf{v} = (0, 1, 2) \).
02

Set Up the Cross Product Determinant

The cross product \( \mathbf{u} \times \mathbf{v} \) can be calculated using the determinant matrix:\[\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \2 & 3 & 0 \0 & 1 & 2 \\end{vmatrix}\]
03

Calculate Cross Product in i-component

Calculate the \( \mathbf{i} \)-component of the cross product by removing the first column and finding the determinant of the remaining 2x2 matrix:\[ \mathbf{i}(3 \cdot 2 - 0 \cdot 1) = \mathbf{i}(6) \].
04

Calculate Cross Product in j-component

Calculate the \( \mathbf{j} \)-component by removing the second column and finding the determinant of the remaining 2x2 matrix:\[ -\mathbf{j}(2 \cdot 2 - 0 \cdot 0) = -\mathbf{j}(4) \].
05

Calculate Cross Product in k-component

Calculate the \( \mathbf{k} \)-component by removing the third column and finding the determinant of the remaining 2x2 matrix:\[ \mathbf{k}(2 \cdot 1 - 3 \cdot 0) = \mathbf{k}(2) \].
06

Write Cross Product in Component Form

Combine the components from the previous steps to express the cross product in vector form:\[ \mathbf{u} \times \mathbf{v} = 6\mathbf{i} - 4\mathbf{j} + 2\mathbf{k} \] or in component form as \( (6, -4, 2) \).
07

Sketch the Vectors

To sketch the vectors, draw \( \mathbf{u} \) from the origin to the point (2, 3, 0) and \( \mathbf{v} \) from the origin to the point (0, 1, 2). Then, draw \( \mathbf{u} \times \mathbf{v} \) starting from the origin to the point (6, -4, 2). Ensure that \( \mathbf{u} \times \mathbf{v} \) is perpendicular to the plane formed by \( \mathbf{u} \) and \( \mathbf{v} \).

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.

Vector Components
Vectors are mathematical entities with both direction and magnitude. A vector can often be broken down into its components, each of which represents the vector's influence in a given direction.
  • In a 3D space, vectors typically have components along the x, y, and z axes.
  • For example, the vector \( \mathbf{u} = 2\mathbf{i} + 3\mathbf{j} \) can be expressed as components \((2, 3, 0)\), where \(\mathbf{i}, \mathbf{j}, \) and \(\mathbf{k}\) represent the unit vectors along the x, y, and z axes respectively.
Understanding vector components is crucial because it serves as the foundation for operations like addition, subtraction, and, notably, the cross product.
Determinant Matrix
A determinant matrix is a special square matrix used in various mathematical calculations, such as finding the cross product of two vectors in three-dimensional space.
  • The formula involves organizing the vectors in a 3x3 matrix with the unit vectors \((\mathbf{i}, \mathbf{j}, \mathbf{k})\) in the first row and the components of the vectors in the next two rows.
  • To calculate the cross product \( \mathbf{u} \times \mathbf{v} \), the determinant of this matrix helps us find the vectors' perpendicular direction.
The formula for the cross product using the determinant matrix is:\[\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \2 & 3 & 0 \0 & 1 & 2 \\end{vmatrix}\] This helps in systematically breaking down the calculation into manageable parts.
3D Vectors
3D vectors are vectors that exist in a three-dimensional space, characterized by their components along the x, y, and z axes.
  • They can describe any quantity that has both magnitude and direction in the real world, like force or velocity.
  • In the exercise, both \(\mathbf{u}\) and \(\mathbf{v}\) are examples of 3D vectors.
The vector \(\mathbf{v}\) consists of components \((0, 1, 2)\), which translates to pointing mainly in the positive y and z directions. Understanding how to manipulate these 3D vectors is essential for applications in physics, engineering, and computer graphics.
Vector Perpendicularity
When two vectors are crossed, the resulting vector is perpendicular to the plane formed by the first two. This is a fundamental property of the cross product.
  • In the given exercise, \( \mathbf{u} \times \mathbf{v} \) is perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \).
  • The cross product is used in fields like physics and engineering to find a perpendicular direction to a given plane, useful in determining torque or angular momentum.
This perpendicularity is not arbitrary but is dictated by the right-hand rule, which is a way to determine the direction of the cross product vector when you place your hand such that your fingers point in the direction of the first vector, and your palm faces the direction of the second. Your outstretched thumb will point in the direction of the perpendicular cross product vector.

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

Use the following figure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems. For the following exercises, the cylindrical coordinates \((r, \theta, z)\) of a point are given. Find the rectangular coordinates \((x, y, z)\) of the point. \((2, \pi,-4)\)

For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates. \(\left(4, \frac{\pi}{4}, \frac{\pi}{6}\right)\)

For the following exercises, consider a small boat crossing a river. An airplane is flying in the direction of \(52^{\circ}\) east of north with a speed of \(450 \mathrm{mph}\). A strong wind has a bearing \(33^{\circ}\) east of north with a speed of \(50 \mathrm{mph}\). What is the resultant ground speed and bearing of the airplane?

For the following exercises, write the given equation in cylindrical coordinates and spherical coordinates. $$ x^{2}+y^{2}+z^{2}=144 $$

[T] Use a CAS to create the intersection between cylinder \(9 x^{2}+4 y^{2}=18 \quad\) and \(\quad\) ellipsoid \(36 x^{2}+16 y^{2}+9 z^{2}=144,\) and find the equations of the intersection curves.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

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.