For the vectors A and B in Fig. E1.24 use the method of components to find the magnitude and direction of the vector difference B - A

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Identify the components of vector A. Since A is along the negative y-axis, its components are: A_x = 0 and A_y = -8.0 m.

Identify the components of vector B. Use trigonometry to find: B_x = 15.0 m * cos(30°) and B_y = 15.0 m * sin(30°).

Calculate the components of the vector difference B - A. Subtract the components of A from B: (B - A)_x = B_x - A_x and (B - A)_y = B_y - A_y.

Determine the magnitude of the vector difference B - A using the Pythagorean theorem: |B - A| = sqrt((B - A)_x^2 + (B - A)_y^2).

Find the direction of the vector difference B - A by calculating the angle θ with respect to the positive x-axis using the tangent function: θ = arctan((B - A)_y / (B - A)_x).

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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 components along the x and y axes. This is done using trigonometric functions: the x-component is found using the cosine of the angle, while the y-component is found using the sine. For example, for a vector A at an angle θ, the components are A_x = A * cos(θ) and A_y = A * sin(θ). Understanding components is essential for performing vector addition or subtraction.

Vector Subtraction

Vector subtraction involves finding the difference between two vectors, which can be achieved by adding the negative of the vector being subtracted. For vectors A and B, the difference B - A can be calculated by determining the components of both vectors and then subtracting the corresponding components: (B_x - A_x, B_y - A_y). This method allows for a clear understanding of the resultant vector's magnitude and direction.

Magnitude and Direction of Vectors

The magnitude of a vector is its length, calculated using the Pythagorean theorem for its components: |V| = √(V_x² + V_y²). The direction is typically expressed as an angle relative to a reference axis, often using the arctangent function: θ = arctan(V_y / V_x). Knowing how to find both the magnitude and direction is crucial for fully describing a vector in a two-dimensional space.