Compute the x- and y-components of the vectors A, B, C, and D in Fig. E1.24.

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Identify the vectors and their respective angles with the positive x-axis: Vector N (14 cm, 13°), Vector M (21 cm, 62°), Vector P (17 cm, 40°), and Vector O (25 cm, 45°).

For each vector, use the trigonometric functions to find the x-component: x-component = magnitude * cos(angle).

For each vector, use the trigonometric functions to find the y-component: y-component = magnitude * sin(angle).

Apply the formulas to Vector N: x-component = 14 * cos(13°), y-component = 14 * sin(13°).

Apply the formulas to Vector M: x-component = 21 * cos(62°), y-component = 21 * sin(62°).

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주요 개념

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

Vector Components

Vectors can be broken down into their x- and y-components, which represent the vector's influence in the horizontal and vertical directions, respectively. This is done using trigonometric functions: the x-component is found using the cosine of the angle, and the y-component is found using the sine of the angle, both multiplied by the vector's magnitude.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are essential for resolving vectors into components. For a vector at an angle θ from the positive x-axis, the cosine function helps find the adjacent side (x-component), while the sine function helps find the opposite side (y-component) of the right triangle formed by the vector.

Reference Angles and Quadrants

Understanding the quadrant in which a vector lies is crucial for determining the signs of its components. Angles are typically measured from the positive x-axis, and the sign of the components depends on the vector's direction: positive in the first quadrant, negative in the second for x, and so on. This helps in correctly applying trigonometric functions to find vector components.