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

Verified step by step guidance

1. Identify the magnitudes and angles of each vector: \(\vec{M} = 21 \text{ cm}, 62^\circ\), \(\vec{N} = 14 \text{ cm}, 13^\circ\), \(\vec{O} = 25 \text{ cm}, 45^\circ\), \(\vec{P} = 17 \text{ cm}, 40^\circ\).
2. For each vector, use the formulas for the x- and y-components: \(A_x = A \cos(\theta)\) and \(A_y = A \sin(\theta)\).
3. Calculate the x-component of each vector: \(M_x = 21 \cos(62^\circ)\), \(N_x = 14 \cos(13^\circ)\), \(O_x = 25 \cos(45^\circ)\), \(P_x = 17 \cos(40^\circ)\).
4. Calculate the y-component of each vector: \(M_y = 21 \sin(62^\circ)\), \(N_y = 14 \sin(13^\circ)\), \(O_y = 25 \sin(45^\circ)\), \(P_y = 17 \sin(40^\circ)\).
5. Combine the results to get the x- and y-components of each vector: \(\vec{M} = (M_x, M_y)\), \(\vec{N} = (N_x, N_y)\), \(\vec{O} = (O_x, O_y)\), \(\vec{P} = (P_x, P_y)\).

Key Concepts

Vector Components

Vector components are the projections of a vector along the axes of a coordinate system, typically the x and y axes in a Cartesian plane. Each vector can be broken down into its horizontal (x) and vertical (y) components using trigonometric functions. For example, if a vector has a magnitude and an angle, the x-component can be found using the cosine of the angle, while the y-component is found using the sine.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the ratios of its sides. In the context of vectors, these functions are essential for calculating the components of a vector based on its angle with respect to the axes. For instance, for a vector at an angle θ, the x-component is given by the product of the vector's magnitude and cos(θ), while the y-component is the product of the magnitude and sin(θ).

Coordinate System

A coordinate system provides a framework for defining the position of points in space using numerical values. In physics, the Cartesian coordinate system is commonly used, where points are defined by their x and y coordinates. Understanding this system is crucial for visualizing vectors and their components, as it allows for the accurate representation of direction and magnitude in two-dimensional space.