/*! 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 18 (I) Calculate the angle between ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 18

(I) Calculate the angle between the vectors: \(\vec { \mathbf { A } } = 6.8 \hat { \mathbf { i } } - 3.4 \hat { \mathbf { j } } - 6.2 \hat { \mathbf { k } } \quad\) and \(\quad \vec { \mathbf { B } } = 8.2 \hat { \mathbf { i } } + 2.3 \hat { \mathbf { j } } - 7.0 \hat { \mathbf { k } }\)

Short Answer

Expert verified
The angle between the vectors is approximately \( 32.21^\circ \).

Step by step solution

01

Write Down the Vectors

We start by defining our vectors. The vector \( \vec{\mathbf{A}} \) is given by: \[ \vec{\mathbf{A}} = 6.8 \hat{\mathbf{i}} - 3.4 \hat{\mathbf{j}} - 6.2 \hat{\mathbf{k}} \] and the vector \( \vec{\mathbf{B}} \) is:\[ \vec{\mathbf{B}} = 8.2 \hat{\mathbf{i}} + 2.3 \hat{\mathbf{j}} - 7.0 \hat{\mathbf{k}} \]
02

Calculate Dot Product of the Vectors

The dot product \( \vec{\mathbf{A}} \cdot \vec{\mathbf{B}} \) is determined by multiplying corresponding components and summing them up:\[ \vec{\mathbf{A}} \cdot \vec{\mathbf{B}} = (6.8)(8.2) + (-3.4)(2.3) + (-6.2)(-7.0) \]Calculating this gives:\[ \vec{\mathbf{A}} \cdot \vec{\mathbf{B}} = 55.76 - 7.82 + 43.4 = 91.34 \]
03

Calculate Magnitude of Each Vector

The magnitude of a vector \( \vec{\mathbf{A}} = (x, y, z) \) is calculated using the formula:\[ |\vec{\mathbf{A}}| = \sqrt{x^2 + y^2 + z^2} \]Magnitude of \( \vec{\mathbf{A}} \):\[ |\vec{\mathbf{A}}| = \sqrt{(6.8)^2 + (-3.4)^2 + (-6.2)^2} = \sqrt{46.24 + 11.56 + 38.44} = \sqrt{96.24} \approx 9.808 \]Magnitude of \( \vec{\mathbf{B}} \):\[ |\vec{\mathbf{B}}| = \sqrt{(8.2)^2 + (2.3)^2 + (-7.0)^2} = \sqrt{67.24 + 5.29 + 49} = \sqrt{121.53} \approx 11.026 \]
04

Use the Dot Product to Find the Angle

The angle \( \theta \) between two vectors can be found using the formula:\[ \cos \theta = \frac{\vec{\mathbf{A}} \cdot \vec{\mathbf{B}}}{|\vec{\mathbf{A}}| \cdot |\vec{\mathbf{B}}|} \]Substitute the values we calculated:\[ \cos \theta = \frac{91.34}{9.808 \times 11.026} \approx \frac{91.34}{108.196} \approx 0.844 \]
05

Calculate the Angle

To find \( \theta \), take the inverse cosine:\[ \theta = \cos^{-1}(0.844) \approx 32.21^\circ \]

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.

Dot Product
The dot product is a fundamental tool in vector mathematics that allows us to determine how "aligned" two vectors are with one another. It is calculated by multiplying the corresponding components of the vectors and summing the results.

In mathematical terms, for two vectors \( \vec{\mathbf{A}} = (a_1, a_2, a_3) \) and \( \vec{\mathbf{B}} = (b_1, b_2, b_3) \), the dot product \( \vec{\mathbf{A}} \cdot \vec{\mathbf{B}} \) is given by:

  • \( \vec{\mathbf{A}} \cdot \vec{\mathbf{B}} = a_1 \cdot b_1 + a_2 \cdot b_2 + a_3 \cdot b_3 \)
The result of this operation is a scalar (a single number), not a vector.

The dot product has several important applications, including:
  • Calculating the angle between vectors
  • Determining the projection of one vector onto another
  • Checking for orthogonality (perpendicularity) 鈥� if the dot product is zero, vectors are perpendicular
Vector Magnitude
Vector magnitude refers to the "length" or "size" of a vector. It is a measure of how long the vector is in a particular space, and it is always a non-negative number.

To calculate the magnitude of a vector \( \vec{\mathbf{A}} = (x, y, z) \), we use the formula:

  • \( |\vec{\mathbf{A}}| = \sqrt{x^2 + y^2 + z^2} \)
This formula is essentially an extension of the Pythagorean theorem to three dimensions, showing the distance from the origin to the point described by the vector.

The magnitude of a vector is crucial in various applications, such as:
  • Normalizing vectors, which involves creating a unit vector (a vector with magnitude 1)
  • Calculating physical quantities like force and velocity in physics
  • Determining distances between points in spatial analyses
Angle Between Vectors
The angle between two vectors is a measure of their direction relative to each other. It is an important concept in vector mathematics, especially when analyzing directional similarity or compatibility between vectors.

To find the angle between two vectors, \( \vec{\mathbf{A}} \) and \( \vec{\mathbf{B}} \), we can utilize the dot product. The formula that helps us is:

  • \( \cos \theta = \frac{\vec{\mathbf{A}} \cdot \vec{\mathbf{B}}}{|\vec{\mathbf{A}}| \cdot |\vec{\mathbf{B}}|} \)
Here, \( \theta \) is the angle between the vectors. To find the angle itself, \( \theta = \cos^{-1}(\cos \theta) \) is used.

Understanding angles between vectors is useful in many fields, including physics, engineering, and computer graphics:
  • It helps in finding relationships between vectors in simulations and animations
  • It assists in optimizing trajectories and directions in engineering applications
  • Helps in detecting similarities or differences in data science vectors

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

A \(265-\mathrm{kg}\) load is lifted \(23.0 \mathrm{~m}\) vertically with an acceleration \(a=0.150 \mathrm{~g}\) by a single cable. Determine \((a)\) the tension in the cable; \((b)\) the net work done on the load; (c) the work done by the cable on the load; \((d)\) the work done by gravity on the load; \((e)\) the final speed of the load assuming it started from rest.

A baseball \((m=145 \mathrm{~g})\) traveling \(32 \mathrm{~m} / \mathrm{s}\) moves a fielder's glove backward \(25 \mathrm{~cm}\) when the ball is caught. What was the average force exerted by the ball on the glove?

A varying force is given by \(F=A e^{-k x},\) where \(x\) is the position; \(A\) and \(k\) are constants that have units of \(\mathrm{N}\) and \(\mathrm{m}^{-1},\) respectively. What is the work done when \(x\) goes from \(0.10 \mathrm{~m}\) to infinity?

A cyclist starts from rest and coasts down a \(4.0^{\circ}\) hill. The mass of the cyclist plus bicycle is \(85 \mathrm{~kg} .\) After the cyclist has traveled \(250 \mathrm{~m},\) (a) what was the net work done by gravity on the cyclist? (b) How fast is the cyclist going? Ignore air resistance.

A \(6.10-\mathrm{kg}\) block is pushed \(9.25 \mathrm{~m}\) up a smooth \(37.0^{\circ}\) inclined plane by a horizontal force of \(75.0 \mathrm{~N}\). If the initial speed of the block is \(3.25 \mathrm{~m} / \mathrm{s}\) up the plane, calculate \((a)\) the initial kinetic energy of the block; (b) the work done by the 75.0-N force; \((c)\) the work done by gravity; \((d)\) the work done by the normal force; \((e)\) the final kinetic energy of the block.
See all solutions

Recommended explanations on Physics Textbooks

Measurements

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Astrophysics

Read Explanation

Classical Mechanics

Read Explanation

Electromagnetism

Read Explanation

Electrostatics

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.