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Chapter 1: Problem 102

Anchored sailboats A sailboat named Ditl is anchored 200 feet north and 300 feet east of an observer standing on shore, while a second sailboat named Windborne is anchored 250 feet north and 100 feet west of the observer. Find the angle between the two sailboats as determined by the observer on shore. GRAPH CANT COPY

Short Answer

Expert verified
Answer: The approximate angle between Ditl and Windborne is 49.10掳.

Step by step solution

01

Determine coordinates of the sailboats

Using the given distance, we can determine the position of each boat in our coordinate system: - Ditl is 200 feet north (y direction) and 300 feet east (x direction), so its coordinates are (300, 200). - Windborne is 250 feet north (y direction) and 100 feet west (x direction), so its coordinates are (-100, 250).
02

Compute the magnitudes of the position vectors

We need to compute the magnitudes of the position vectors, which are the distances between the observer (origin) and the two sailboats: Magnitude of the position vector for Ditl: \(||\vec{D}|| = \sqrt{(300)^2 + (200)^2} = \sqrt{130000} = 100\sqrt{13}\) Magnitude of the position vector for Windborne: \(||\vec{W}|| = \sqrt{(-100)^2 + (250)^2} = \sqrt{72500} = 25\sqrt{29}\)
03

Calculate the dot product of the position vectors

Now let's calculate the dot product of the position vectors for Ditl \(\vec{D}\) and Windborne \(\vec{W}\): \(\vec{D} \cdot \vec{W} = (300)(-100) + (200)(250) = -30000 + 50000 = 20000\)
04

Find the cosine of the angle between the sailboats

Using the magnitudes of the position vectors and the dot product, we can find the cosine of the angle between the two sailboats (\(\theta\)) using the formula: \(\cos{\theta} = \frac{\vec{D} \cdot \vec{W}}{||\vec{D}|| \cdot ||\vec{W}||} = \frac{20000}{100\sqrt{13} \cdot 25\sqrt{29}} = \frac{8}{\sqrt{13} \cdot \sqrt{29}}\)
05

Calculate the angle between the sailboats

To find the angle between the sailboats, take the inverse cosine of the calculated cosine value: \(\theta = \cos^{-1}{\left(\frac{8}{\sqrt{13} \cdot \sqrt{29}}\right)} \approx 49.10掳\) So, the angle between two sailboats, as determined by the observer on shore, is approximately 49.10掳.

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Key Concepts

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

Dot Product
The dot product, also known as the scalar product, is a fundamental operation in vector calculus. It's a way to multiply two vectors such that the result is a scalar (a simple number).
The formula for finding the dot product of two vectors, say \( \vec{A} \) and \( \vec{B} \), is given by:
  • \( \vec{A} \cdot \vec{B} = A_xB_x + A_yB_y \)
  • It extends to n-dimensions as \( A_1B_1 + A_2B_2 + \ldots + A_nB_n \)
For the exercise, the position vectors of the sailboats are given in coordinates.
Ditl, say \( \vec{D} \), and Windborne, say \( \vec{W} \), interact through this calculation:
  • \( \vec{D} = (300, 200) \)
  • \( \vec{W} = (-100, 250) \)
  • The dot product \( \vec{D} \cdot \vec{W} = (300)(-100) + (200)(250) = -30000 + 50000 = 20000 \)
This result is essential as it provides insight when calculating angles between vectors later.
Angle Calculation
Calculating the angle between two vectors involves several steps, leveraging the dot product as a key component.
To find the angle \( \theta \) between two vectors \( \vec{A} \) and \( \vec{B} \), you can use:
  • \( \cos{\theta} = \frac{\vec{A} \cdot \vec{B}}{||\vec{A}|| \cdot ||\vec{B}||} \)
This formula tells us that the cosine of the angle is the ratio of the dot product to the product of the magnitudes of the vectors.
In our scenario, with Ditl and Windborne:
  • First, determine the magnitudes as \( ||\vec{D}|| = 100\sqrt{13} \) and \( ||\vec{W}|| = 25\sqrt{29} \).
  • Then find \( \cos{\theta} = \frac{20000}{100\sqrt{13} \cdot 25\sqrt{29}} = \frac{8}{\sqrt{13} \cdot \sqrt{29}} \).
  • Finally, determine \( \theta \) using \( \theta = \cos^{-1}{\left(\frac{8}{\sqrt{13} \cdot \sqrt{29}}\right)} \).
This calculation ultimately provides the angle between two vectors, in this case, approximately 49.10掳, helping us understand their spatial relationship.
Position Vectors
Position vectors are extremely useful when describing the location of points relative to a fixed origin.
In 2D space, these vectors are often represented with coordinates (x, y) from the origin at point (0,0).
For our problem, the observer represents the origin, and the sailboats Ditl and Windborne have specific position vectors:
  • Ditl: Located at 200 feet north and 300 feet east, represented as \( \vec{D} = (300, 200) \).
  • Windborne: Positioned at 250 feet north and 100 feet west, given as \( \vec{W} = (-100, 250) \).
These vectors define the sailboats' positions as points in the plane and serve as the basis for subsequent calculations, such as determining vector magnitudes and finding the angle between them.
Position vectors simplify complex spatial problems by converting physical locations into numeric data easily utilized in mathematical operations.

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