For the two vectors A→\overrightarrow{A} and B→\overrightarrow{B} in the figure1.391.39, find the magnitude and direction of the vector product A→×B→\overrightarrow{A}\times\overrightarrow{B}.

Ch 01: Units, Physical Quantities & Vectors

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

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Young & Freedman Calc 14th Edition

Ch 01: Units, Physical Quantities & Vectors

Problem 48b

Chapter 1, Problem 48b

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

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Identify the components of vectors A and B. Vector A has a magnitude of 3.2 m and is directed at an angle of 28° below the negative x-axis. Vector B has a magnitude of 4.2 m and is directed at an angle of 52° above the positive x-axis.

Calculate the components of vector A. The x-component of A is A_x = 3.2 * cos(28°) and the y-component is A_y = 3.2 * sin(28°). Since A is directed below the x-axis, A_y will be negative.

Calculate the components of vector B. The x-component of B is B_x = 4.2 * cos(52°) and the y-component is B_y = 4.2 * sin(52°).

Use the formula for the vector product (cross product) A x B = |A| * |B| * sin(θ) * n, where θ is the angle between A and B, and n is the unit vector perpendicular to the plane containing A and B. The angle θ can be found by adding the angles of A and B with respect to the x-axis, which is 28° + 52° = 80°.

Calculate the magnitude of the cross product using the formula |A x B| = |A| * |B| * sin(80°). The direction of the cross product is determined by the right-hand rule, which will be perpendicular to the plane containing A and B.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vector Product (Cross Product)

The vector product, or cross product, of two vectors results in a third vector that is perpendicular to the plane containing the original vectors. Its magnitude is given by |A x B| = |A||B|sin(θ), where θ is the angle between vectors A and B. The direction follows the right-hand rule, which helps determine the orientation of the resulting vector.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem in a Cartesian plane. For a vector with components (x, y), the magnitude is √(x² + y²). In the context of the cross product, the magnitude of the resulting vector depends on the sine of the angle between the original vectors and their individual magnitudes.

Direction of a Vector

The direction of a vector is defined by the angle it makes with a reference axis, typically the x-axis in a Cartesian coordinate system. For the cross product, the direction is perpendicular to the plane formed by the two vectors, determined by the right-hand rule. This concept is crucial for understanding how the cross product vector is oriented in space.