Textbook Question
Use the figure to find each vector: - v. Use vector notation as in Example 4.
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8. Vectors
Geometric Vectors
Problem 7.54
Textbook Question
A plane flies 650 mph on a bearing of 175.3°. A 25-mph wind, from a direction of 266.6°, blows against the plane. Find the resulting bearing of the plane.
Verified step by step guidance1
Convert the plane's bearing and the wind's direction from degrees to radians for calculation purposes. Use the formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).
Determine the components of the plane's velocity vector and the wind's velocity vector. Use the formulas \( V_x = V \cos(\theta) \) and \( V_y = V \sin(\theta) \), where \( V \) is the speed and \( \theta \) is the angle in radians.
Calculate the resultant velocity vector by adding the components of the plane's velocity vector and the wind's velocity vector. Use \( V_{rx} = V_{px} + V_{wx} \) and \( V_{ry} = V_{py} + V_{wy} \).
Compute the magnitude of the resultant velocity vector using the Pythagorean theorem: \( V_r = \sqrt{V_{rx}^2 + V_{ry}^2} \).
Find the resultant bearing of the plane by calculating the angle of the resultant velocity vector. Use the formula \( \theta_r = \tan^{-1}(\frac{V_{ry}}{V_{rx}}) \) and convert the angle from radians back to degrees.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Bearing
Bearing is a way of describing direction in navigation, measured in degrees from North (0°) clockwise. For example, a bearing of 175.3° indicates a direction slightly east of due South. Understanding bearings is crucial for solving problems involving navigation and movement in a two-dimensional plane.
Vector Addition
Vector addition involves combining two or more vectors to determine a resultant vector. In this context, the plane's velocity and the wind's velocity are represented as vectors. The resultant vector will give the new direction and speed of the plane, which is essential for finding the resulting bearing.
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Trigonometric Functions
Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. These functions are used to resolve the components of the vectors (plane and wind) into their respective horizontal and vertical components, facilitating the calculation of the resultant vector's direction and magnitude.
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