/*! 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 166 Determine the direction cosines ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 166

Determine the direction cosines of vector \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}+2 \mathbf{k} \quad\) and \(\quad\) show \(\quad\) they \(\quad\) satisfy \(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1\)

Short Answer

Expert verified
The direction cosines are \(\frac{1}{3}\), \(\frac{2}{3}\), \(\frac{2}{3}\) and they satisfy \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\).

Step by step solution

01

Calculate the Magnitude of the Vector

To find the direction cosines, we first calculate the magnitude of vector \( \mathbf{u} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k} \). The magnitude is given by the formula: \[ \| \mathbf{u} \| = \sqrt{(1)^2 + (2)^2 + (2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]
02

Find Direction Cosines

The direction cosines are the cosines of the angles that the vector makes with the coordinate axes. They are calculated as follows: \( \cos \alpha = \frac{1}{3} \), \( \cos \beta = \frac{2}{3} \), \( \cos \gamma = \frac{2}{3} \).
03

Verify the Direction Cosines Condition

To verify they satisfy the condition, substitute the values into the equation: \[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = \left(\frac{1}{3}\right)^2 + \left(\frac{2}{3}\right)^2 + \left(\frac{2}{3}\right)^2 \] Simplifying, we get: \[ \frac{1}{9} + \frac{4}{9} + \frac{4}{9} = \frac{9}{9} = 1 \] This shows that the condition is satisfied.

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 Magnitude
When we talk about a vector's magnitude, we are referring to its length or size. It's a measure of how long the vector is when drawn as an arrow in space. To find the magnitude of a vector, we use the Pythagorean theorem extended into three dimensions. For example, consider the vector \( \mathbf{u} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k} \). To calculate its magnitude, we sum the squares of its components and then take the square root:
  • Step 1: Square each component: \( 1^2 = 1 \), \( 2^2 = 4 \), and \( 2^2 = 4 \).
  • Step 2: Add these squares together: \( 1 + 4 + 4 = 9 \).
  • Step 3: Take the square root of the total: \( \sqrt{9} = 3 \).
This result gives us a magnitude of 3 for vector \( \mathbf{u} \). Magnitude is crucial because it influences how we calculate direction cosines and express vectors in other forms like unit vectors.
Coordinate Axes
Coordinate axes are the foundation of the Cartesian coordinate system, which allows us to locate and describe points and vectors in three-dimensional space. These axes are imaginary lines: the x, y, and z-axes, that intersect at a single point called the origin.Vectors are expressed in terms of their components along each axis. For instance, in the vector \( \mathbf{u} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k} \), the coefficients represent how far the vector extends along the x, y, and z-axes respectively:
  • \( \mathbf{i} \) represents one unit along the x-axis.
  • 2\( \mathbf{j} \) represents two units along the y-axis.
  • 2\( \mathbf{k} \) represents two units along the z-axis.
These components help us visualize the vector in three-dimensional space, making it easier to analyze its direction and magnitude.
Angle Cosines
Angle cosines, also known as direction cosines, describe how a vector is oriented with respect to the coordinate axes. Imagine a vector in space making angles with each of the three axes. The cosines of these angles are called the direction cosines.For the vector \( \mathbf{u} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k} \), the direction cosines are calculated as follows:
  • First, divide each component of the vector by its magnitude. This converts the vector to a unit vector pointing in the same direction.
  • For \( \mathbf{u} \), with a magnitude of 3:
    • \( \cos \alpha = \frac{1}{3} \)
    • \( \cos \beta = \frac{2}{3} \)
    • \( \cos \gamma = \frac{2}{3} \)
These values tell us how the vector aligns with each axis. The property that \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \) is always true for vectors because it relates to the Pythagorean identity in three dimensions, confirming that the vector鈥檚 length, direction, and the three angles are consistent.
Mathematics Problem Solving
Solving mathematics problems often involves a systematic approach to unravel complex scenarios using fundamental concepts. These principles build on each other, helping students grasp the more advanced topics.Let's approach the problem of finding direction cosines systematically:
  • Identify what is given and what needs to be found. In this example, you first determine the magnitude of vector \( \mathbf{u} \).
  • Use known formulas, such as those for vector magnitude and trigonometric identities, to break down the problem. Analyze each part step-by-step, such as computing for direction cosines.
  • Verify your results. After calculations, check if they satisfy the mathematical conditions given, like verifying \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \).
This structured method not only solves problems but also deepens understanding and ensures accuracy by covering all bases from start to end, making it approachable and digestible for learners.

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

For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle \(\varphi\) in radians rounded to four decimal places. \(\left(3, \frac{\pi}{2}, 3\right)\)

Hyperboloid of one sheet \(25 x^{2}+25 y^{2}-z^{2}=25\) and elliptic cone \(-25 x^{2}+75 y^{2}+z^{2}=0\) are represented in the following figure along with their intersection curves. Identify the intersection curves and find their equations (Hint: Find \(y\) from the system consisting of the equations of the surfaces.)

Determine the intersection points of elliptic cone \(x^{2}-y^{2}-z^{2}=0\) with the line of symmetric equations \(\frac{x-1}{2}=\frac{y+1}{3}=z\)

For the following exercises, find the most suitable system of coordinates to describe the solids. The solid situated in the first octant with a vertex at the origin and enclosed by a cube of edge length \(a,\) where \(a>0\)

\(\begin{array}{llll} & \text { Show } & \text { that } & \text { quadric }\end{array}\) surface \(x^{2}+y^{2}+z^{2}+2 x y+2 x z+2 y z+x+y+z=0\) reduces to two parallel planes.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.