/*! 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 14 In Exercises \(9-14,\) sketch th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 14

In Exercises \(9-14,\) sketch the coordinate axes and then include the vectors \(\mathbf{u}, \mathbf{v}\) and \(\mathbf{u} \times \mathbf{v}\) as vectors starting at the origin. $$ \mathbf{u}=\mathbf{j}+2 \mathbf{k}, \quad \mathbf{v}=\mathbf{i} $$

Short Answer

Expert verified
Draw \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{u} \times \mathbf{v} \) on the coordinate axes based on their components.

Step by step solution

01

Understand Vector Components

The vector \( \mathbf{u} = \mathbf{j} + 2\mathbf{k} \) means that it has no component in the x-direction, 1 component in the y-direction, and 2 components in the z-direction. The vector \( \mathbf{v} = \mathbf{i} \) means that it has 1 component in the x-direction, and no components in the y- or z-directions.
02

Draw Vector u

On the 3D coordinate axes, start at the origin and draw the vector \( \mathbf{u} \). Since \( \mathbf{u} = \mathbf{j} + 2\mathbf{k} \), draw a vector 1 unit in the positive y-direction and 2 units in the positive z-direction.
03

Draw Vector v

On the same coordinate axes, start at the origin and draw the vector \( \mathbf{v} \). Since \( \mathbf{v} = \mathbf{i} \), draw a vector 1 unit in the positive x-direction.
04

Compute Cross Product u 脳 v

Use the formula for cross product: \( \mathbf{u} \times \mathbf{v} = (a_2b_3 - a_3b_2)\mathbf{i} + (a_3b_1 - a_1b_3)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k} \) where \( \mathbf{u} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \) and \( \mathbf{v} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \). Here, \( \mathbf{u} = 0\mathbf{i} + 1\mathbf{j} + 2\mathbf{k} \) and \( \mathbf{v} = 1\mathbf{i} + 0\mathbf{j} + 0\mathbf{k} \). Simplifying, we get \( \mathbf{u} \times \mathbf{v} = 0\mathbf{i} + 2\mathbf{j} - 1\mathbf{k} \).
05

Draw Cross Product u 脳 v

Draw the vector \( \mathbf{u} \times \mathbf{v} = 2\mathbf{j} - \mathbf{k} \) starting at the origin. This means a vector 2 units in the positive y-direction and 1 unit in the negative z-direction.

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.

Cross Product
The cross product is a mathematical operation used to find a vector that is perpendicular to two given vectors in three-dimensional space. If you have two vectors \( \mathbf{u} \) and \( \mathbf{v} \), their cross product is denoted as \( \mathbf{u} \times \mathbf{v} \). This operation is particularly useful in physics and engineering because it helps to determine orientation and rotational effects.
  • Formula: The formula for the cross product of two vectors \( \mathbf{u} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \) and \( \mathbf{v} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k} \) is given by: \[ \mathbf{u} \times \mathbf{v} = (a_2b_3 - a_3b_2)\mathbf{i} + (a_3b_1 - a_1b_3)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k} \]
  • Components: Notice how each component (\( \mathbf{i}, \mathbf{j}, \mathbf{k} \)) of the resulting vector derives from interactions of two components at a time,effectively using a determinant from linear algebra to calculate each substantial result.
  • Result: The resulting vector from the cross product always points in a direction perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \).
Remember, the magnitude of the cross product is given by the area of the parallelogram that the two vectors span, calculated as \( |\mathbf{u} \times \mathbf{v}| = |\mathbf{u}| |\mathbf{v}| \sin \theta \), where \( \theta \) is the angle between \( \mathbf{u} \) and \( \mathbf{v} \).
3D Coordinate System
Understanding the 3D coordinate system is essential for working with vectors in spatial analysis. This system is composed of three axes: the x-axis, y-axis, and z-axis. These axes are perpendicular to each other and intersect at a point called the origin, often labeled as \((0, 0, 0)\).
  • X-Axis: Extends horizontally, where positive values move to the right and negative to the left.
  • Y-Axis: Extends vertically, where positive values move upwards and negative downwards.
  • Z-Axis: Represents depth, coming out of or going into the page. Positive values move out towards the observer, and negative values move away.
A point or vector in this space can be described with coordinates \((x, y, z)\). For vectors like \( \mathbf{u} \) and \( \mathbf{v} \) in the original problem, their position and direction are described with reference to these three axes.Visualizing this system in your mind or drawing it helps in understanding operations like the cross product and ensuring the correct placement of vectors.
Vector Components
Vector components in three-dimensional space describe how a vector can be broken down along the x, y, and z axes. Each vector is a combination of these directions.
  • X-Component: Describes the length of the vector in the direction of the x-axis, often denoted as a multiple of \( \mathbf{i} \).
  • Y-Component: Shows the vector's length along the y-axis and is expressed in terms of the unit vector \( \mathbf{j} \).
  • Z-Component: Represents how far the vector extends in the z-direction, multiplied by \( \mathbf{k} \).
The vector \( \mathbf{u} = \mathbf{j} + 2 \mathbf{k} \) from the problem highlights that it has no x-component, a y-component of 1, and a z-component of 2. Similarly, \( \mathbf{v} = \mathbf{i} \) only has an x-component and lacks components in the y and z directions.Visualizing these components helps in both computation and geometrical representation, ensuring you're able to manipulate vectors effectively in applied problems.

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

Suppose you set \(z=0\) in the equation $$ \begin{array}{c}{A x^{2}+B y^{2}+C z^{2}+D x y+E y z+} \\ {F x z+G x+H y+J z+K=0}\end{array} $$ to obtain a curve in the \(x y\) -plane. What will the curve be like? Give reasons for your answer.

Cancellation in cross products If \(\mathbf{u} \times \mathbf{v}=\mathbf{u} \times \mathbf{w}\) and \(\mathbf{u} \neq \mathbf{0}\) then does \(\mathbf{v}=\mathbf{w} ?\) Give reasons for your answer.

Let \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be vectors. Which of the following make sense, and which do not? Give reasons for your answers. $$ \begin{array}{ll}{\text { a. }(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}} & {\text { b. } \mathbf{u} \times(\mathbf{v} \cdot \mathbf{w})} \\\ {\mathbf{c} . \mathbf{u} \times(\mathbf{v} \times \mathbf{w})} & {\text { d. } \mathbf{u} \cdot(\mathbf{v} \cdot \mathbf{w})}\end{array} $$

How can you tell when two planes \(A_{1} x+B_{1} y+C_{1} z=D_{1}\) and \(A_{2} x+B_{2} y+C_{2} z=D_{2}\) are parallel? Perpendicular? Give reasons for your answer.

Inclined plane How much work does it take to slide a crate 20 \(\mathrm{m}\) along a loading dock by pulling on it with a 200 \(\mathrm{N}\) force at an angle of \(30^{\circ}\) from the horizontal?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Statistics

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.