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

검증된 단계별 안내

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.

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주요 개념

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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.

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.

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.