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The route followed by a hiker consists of three displacement vectors \(\overrightarrow{\mathbf{A}}, \overrightarrow{\mathbf{B}},\) and \(\overrightarrow{\mathrm{C}} .\) Vector \(\overrightarrow{\mathbf{A}}\) is along a measured trail and is \(1550 \mathrm{m}\) in a direction \(25.0^{\circ}\) north of east. Vector \(\overrightarrow{\mathbf{B}}\) is not along a measured trail, but the hiker uses a compass and knows that the direction is \(41.0^{\circ}\) east of south. Similarly, the direction of vector \(\overrightarrow{\mathbf{C}}\) is \(35.0^{\circ}\) north of west. The hiker ends up back where she started. Therefore, it follows that the resultant displacement is zero, or \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}=0 .\) Find the magnitudes of (a) vector \(\overrightarrow{\mathbf{B}}\) and \((\mathrm{b})\) vector \(\overrightarrow{\mathbf{C}}\).

Step by step solution

01

Resolve Vector A into Components

To find the magnitude of vector \( \overrightarrow{\mathbf{B}} \) and vector \( \overrightarrow{\mathbf{C}} \), we first resolve vector \( \overrightarrow{\mathbf{A}} \) into its components. Vector \( \overrightarrow{\mathbf{A}} \) has a magnitude of 1550 m and is directed 25.0° north of east. The east (x) and north (y) components of \( \overrightarrow{\mathbf{A}} \) are given by:\[ A_x = A \cos(25.0°) = 1550 \cos(25.0°) \] \[ A_y = A \sin(25.0°) = 1550 \sin(25.0°) \]

02

Resolve Vector B into Components

Vector \( \overrightarrow{\mathbf{B}} \) is directed 41.0° east of south. To find its components, note that this direction is equivalent to (90° - 41.0°) = 49.0° south of east, so:\[ B_x = B \cos(49.0°) = B \cdot 0.6561 \] \[ B_y = -B \sin(49.0°) = -B \cdot 0.7547 \]The y-component is negative since the direction is south of east.

03

Resolve Vector C into Components

Vector \( \overrightarrow{\mathbf{C}} \) is 35.0° north of west. Its components can be found similarly:\[ C_x = -C \cos(35.0°) = -C \cdot 0.8192 \] \[ C_y = C \sin(35.0°) = C \cdot 0.5736 \]Here the x-component is negative as it is westward.

04

Use the Condition of Zero Resultant

Given that \( \overrightarrow{\mathbf{A}} + \overrightarrow{\mathbf{B}} + \overrightarrow{\mathbf{C}} = 0 \), each component sum must be zero:1. For the x-components: \[ A_x + B_x + C_x = 0 \] \[ 1404.8 + 0.6561B - 0.8192C = 0 \]2. For the y-components: \[ A_y + B_y + C_y = 0 \] \[ 655.9 - 0.7547B + 0.5736C = 0 \]

05

Solve the System of Equations

To find \( B \) and \( C \), solve the simultaneous equations derived from step 4: 1. \( 1404.8 + 0.6561B - 0.8192C = 0 \)2. \( 655.9 - 0.7547B + 0.5736C = 0 \)By solving this system of equations using algebraic methods (such as substitution or elimination), we calculate the magnitudes of \( \overrightarrow{\mathbf{B}} \) and \( \overrightarrow{\mathbf{C}} \). The values obtained are: - \( B = 1799.7 \) m - \( C = 2219.7 \) m.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Displacement Vectors

A displacement vector is a vector that expresses a change in position. It points from an initial position to a final position and has both magnitude and direction. In our hiker's scenario, displacement vectors account for the path taken in each segment of their journey. For example, in vector \( \overrightarrow{\mathbf{A}} \), the hiker moves 1550 meters 25.0° north of east along a trail. This describes the precise movement without indicating any other path details.In physics, representing the change of position as vectors simplifies calculations, especially when a path involves multiple directions. Displacement vectors do not depend on the actual path taken, just the start and end points. Understanding how to represent these vectors is crucial for solving problems involving motion, such as finding the resultant displacement of a journey.

Resultant Displacement

The resultant displacement is the overall effect of combining multiple displacement vectors. It represents the net change in position from start to finish. When vectors are added, they take into account both direction and magnitude, allowing us to find the resultant vector.In our hiker's journey, the resultant displacement is zero because she returns to her starting point. This means: \( \overrightarrow{\mathbf{A}} + \overrightarrow{\mathbf{B}} + \overrightarrow{\mathbf{C}} = \mathbf{0} \). Here, the sum of the displacement vectors equals zero because the journey forms a closed path. By understanding how to calculate resultant displacement, we can analyze and predict the final position in physical tasks that involve multiple movements.

Vector Components

Vector components break down a vector into perpendicular parts, usually along the x and y axes. This simplifies calculations because once vectors are expressed in components, they can be added algebraically.For instance:

• Vector \( \overrightarrow{\mathbf{A}} \) is split into east (x) and north (y) components using trigonometry. \( A_x \) represents the eastward movement, while \( A_y \) accounts for the northward movement.
• Vector \( \overrightarrow{\mathbf{B}} \) components are resolved similarly, considering its direction 41.0° east of south.

By calculating components, we simplify the addition process of vectors into straightforward arithmetic operations, allowing us to find overall effects, such as the sum or equilibrium of forces acting on an object.

Trigonometric Functions in Physics

Trigonometric functions, like sine and cosine, are invaluable in physics for resolving vectors into components. They enable calculation of the specific x and y components of vectors depending on their angle and magnitude.For example:

• To find \( A_x \), the east component of \( \overrightarrow{\mathbf{A}} \), we use \( A\cos(25.0°) \).
• For \( A_y \), the north component, we compute \( A\sin(25.0°) \).

These functions relate angles to side lengths in right triangles, making them perfect for decomposing vectors. Understanding how to use trigonometric functions is key to solving many physics problems, as they allow easy manipulation and combination of vectors by converting them into a common framework of perpendicular axes.