Textbook Question
Starting at point A, a ship sails 18.5 km on a bearing of 189°, then turns and sails 47.8 km on a bearing of 317°. Find the distance of the ship from point A.
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8. Vectors
Geometric Vectors
Problem 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
Identify the velocity vectors of the plane and the wind. The plane's velocity vector has a magnitude of 650 mph and a direction (bearing) of 175.3°. The wind's velocity vector has a magnitude of 25 mph and a direction of 266.6° (wind direction means the wind is coming from this bearing, so the wind's velocity vector points toward 266.6° + 180° = 86.6°).
Convert both velocity vectors from polar form (magnitude and bearing) to Cartesian coordinates (x and y components) using the formulas: \(x = v \times \sin(\theta)\) and \(y = v \times \cos(\theta)\), where \(\theta\) is the bearing angle in degrees.
Add the corresponding components of the plane's velocity vector and the wind's velocity vector to find the resultant velocity vector components: \(V_x = V_{plane,x} + V_{wind,x}\) and \(V_y = V_{plane,y} + V_{wind,y}\).
Calculate the magnitude of the resultant velocity vector using the Pythagorean theorem: \(V = \sqrt{V_x^2 + V_y^2}\) (this step is optional if only the bearing is required).
Find the resulting bearing of the plane by calculating the angle of the resultant vector relative to the north direction using the inverse tangent function: \(\theta = \arctan\left(\frac{V_x}{V_y}\right)\). Adjust the angle to the correct quadrant and convert it to a bearing between 0° and 360°.
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Addition in Navigation
In navigation problems, the actual path of an object is found by adding its velocity vector to the velocity vector of external factors like wind. This involves breaking velocities into components, adding them, and then recombining to find the resultant vector representing the true direction and speed.
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Bearing and Angle Measurement
Bearing is a way to express direction using degrees measured clockwise from the north (0° or 360°). Understanding how to interpret and convert bearings into standard angles for calculations is essential for solving problems involving directions and navigation.
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Trigonometric Components of Vectors
Vectors can be decomposed into horizontal (x) and vertical (y) components using sine and cosine functions based on their angles. This decomposition allows for precise calculation of resultant vectors by summing components along each axis before finding magnitude and direction.
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