Draw each of the following vectors. Then find its x- and y-components. F = (50.0 N, 36.9 degrees counterclockwise from the positive y-axis)

검증된 단계별 안내

Step 1: Understand the problem. The vector F has a magnitude of 50.0 N and is oriented at an angle of 36.9 degrees counterclockwise from the positive y-axis. We need to find its x- and y-components.

Step 2: Recognize the coordinate system. Since the angle is given counterclockwise from the positive y-axis, we need to adjust the angle to fit the standard convention where angles are measured counterclockwise from the positive x-axis. The adjusted angle is θ = 90° - 36.9°.

Step 3: Use trigonometric functions to find the components. The x-component of the vector is given by Fₓ = F * cos(θ), and the y-component is given by Fᵧ = F * sin(θ). Substitute the magnitude of F and the adjusted angle θ into these formulas.

Step 4: Write the expressions for the components. Using the formulas: Fₓ = 50.0 * cos(90° - 36.9°) and Fᵧ = 50.0 * sin(90° - 36.9°). Simplify the trigonometric expressions if needed.

Step 5: Conclude the process. Once the trigonometric calculations are performed, the x- and y-components of the vector F will be determined. Ensure the signs of the components are consistent with the vector's direction in the coordinate system.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Vector Representation

Vectors are quantities that have both magnitude and direction. In this context, the force vector F is represented by its magnitude (50.0 N) and its direction (36.9 degrees counterclockwise from the positive y-axis). Understanding how to graphically represent vectors is essential for visualizing their components.

Components of a Vector

The components of a vector are its projections along the axes of a coordinate system, typically the x-axis and y-axis. For a vector given in polar form, such as F, the x-component can be found using F_x = F * cos(θ) and the y-component using F_y = F * sin(θ), where θ is the angle from the positive x-axis.

Trigonometric Functions

Trigonometric functions, specifically sine and cosine, are fundamental in resolving vectors into their components. These functions relate the angles of a triangle to the ratios of its sides, allowing us to calculate the x- and y-components of a vector based on its angle and magnitude. Mastery of these functions is crucial for solving problems involving vector decomposition.