For the vectors A and B in Fig. E1.24, use a scale drawing to find the magnitude and direction of the vector difference A − B.

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Identify the components of vector A. Since vector 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. Vector B makes a 30-degree angle with the positive y-axis. Calculate its components using trigonometry: B_x = 15.0 m * sin(30°) and B_y = 15.0 m * cos(30°).

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

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

Find the direction of the vector difference A - B. Use the arctangent function to find the angle θ with respect to the x-axis: θ = arctan((A - B)_y / (A - B)_x). Adjust the angle based on the quadrant in which the vector lies.

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Vector Addition and Subtraction

Vector addition and subtraction involve combining vectors to find a resultant vector. When subtracting vector B from vector A (A - B), it can be visualized as adding vector A to the negative of vector B. This requires understanding both the magnitude and direction of the vectors involved, as they are represented in a Cartesian coordinate system.

Magnitude and Direction of Vectors

The magnitude of a vector is its length, representing the quantity it describes, while the direction indicates where the vector points in space. In the context of vector operations, both magnitude and direction must be considered to accurately determine the resultant vector's characteristics after performing operations like addition or subtraction.

Scale Drawing Techniques

Scale drawing techniques involve representing vectors graphically to visualize their relationships. By using a consistent scale, one can draw vectors to scale and measure their lengths and angles accurately. This method is particularly useful for solving vector problems, as it allows for a clear representation of vector addition or subtraction through geometric means.