Ch 01: Units, Physical Quantities & Vectors

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo

Young & Freedman Calc 14th Edition

Ch 01: Units, Physical Quantities & Vectors

Problema 46a

Capitolo 1, Problema 46a

For the two vectors in Fig. E1.35, find the magnitude and direction of the vector product A x B

Guida verificata passo dopo passo

First, understand that the vector product, also known as the cross product, of two vectors $ \mathbf{A} $ and $ \mathbf{B} $ is given by $ \mathbf{A} \times \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \sin(\theta) \mathbf{n} $, where $ \theta $ is the angle between the vectors and $ \mathbf{n} $ is a unit vector perpendicular to the plane containing $ \mathbf{A} $ and $ \mathbf{B} $.

Calculate the magnitudes of vectors $ \mathbf{A} $ and $ \mathbf{B} $ from the given lengths: $ |\mathbf{A}| = 2.80 \text{ cm} $ and $ |\mathbf{B}| = 1.90 \text{ cm} $.

Determine the angle $ \theta $ between the vectors $ \mathbf{A} $ and $ \mathbf{B} $. From the diagram, both vectors make an angle of $ 60.0^\circ $ with the x-axis, so the angle between them is $ 120.0^\circ $.

Substitute the values into the cross product formula: $ |\mathbf{A} \times \mathbf{B}| = (2.80 \text{ cm})(1.90 \text{ cm}) \sin(120.0^\circ) $. Use the fact that $ \sin(120.0^\circ) = \sin(60.0^\circ) = \frac{\sqrt{3}}{2} $.

The direction of the vector product $ \mathbf{A} \times \mathbf{B} $ is given by the right-hand rule. Point your fingers in the direction of $ \mathbf{A} $ and curl them towards $ \mathbf{B} $; your thumb will point in the direction of $ \mathbf{A} \times \mathbf{B} $, which is perpendicular to the plane containing $ \mathbf{A} $ and $ \mathbf{B} $. In this case, it will be out of the page.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Vector Product (Cross Product)

The vector product, or cross product, of two vectors A and B, denoted as A x B, results in a third vector that is perpendicular to the plane formed by A and B. The magnitude of the cross product is given by |A||B|sin(θ), where θ is the angle between the two vectors. This operation is essential in physics for determining torque, angular momentum, and the magnetic force on a charged particle.

Video consigliato:

Percorso guidato

10:30

Vector (Cross) Product and the Right-Hand-Rule

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem in Cartesian coordinates. For a vector represented in two dimensions, the magnitude can be found using the formula |A| = √(Ax² + Ay²), where Ax and Ay are the vector's components along the x and y axes. Understanding the magnitude is crucial for calculating the vector product and interpreting physical quantities.

Video consigliato:

Percorso guidato

03:59

Calculating Magnitude & Components of a Vector

Direction of a Vector

The direction of a vector indicates the orientation of the vector in space and is often expressed in terms of angles relative to a reference axis. In the context of the vector product, the direction of the resulting vector A x B is determined by the right-hand rule, which states that if the fingers of the right hand curl from vector A to vector B, the thumb points in the direction of A x B. This concept is vital for visualizing and solving problems involving vector interactions.

Video consigliato:

Percorso guidato

06:44

Adding 3 Vectors in Unit Vector Notation

Pratica correlata

Domanda del libro di testo

For the two vectors $\vec{A}$ and $\vec{B}$ in the figure $1.39$ , find the magnitude and direction of the vector product $\vec{A} \times \vec{B}$ .

2218

views

Domanda del libro di testo

Find the vector product A x B (expressed in unit vectors) of the two vectors given in Exercise 1.38. What is the magnitude of the vector product? Given two vectors A = 4.00 i + 7.00j and B = 5.00 i − 2.00

2369

views

Domanda del libro di testo

For the two vectors A and B in Fig. E1.39, find the scalar product A · B

2275

views

rank

Domanda del libro di testo

Given two vectors A = -2i + 3j + 4k and B = 3.00î + 1.00ĵ − 3.00k, find the magnitude of each vector.

3094

views

Domanda del libro di testo

Find the scalar product of the vectors A and B given in Exercise 1.38.

1915

views