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

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Young & Freedman Calc 14th Edition

Ch 01: Units, Physical Quantities & Vectors

문제 48b

1장, 문제 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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주요 개념

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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.