Textbook Question
The bearing of a lighthouse from a ship was found to be N 37° E. After the ship sailed 2.5 mi due south, the new bearing was N 25° E. Find the distance between the ship and the lighthouse at each location.
Ch. 7 - Applications of Trigonometry and Vectors
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 8, Problem 36
Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
-5v
Verified step by step guidance1
Identify the vector \( \mathbf{v} = \langle 4, 3 \rangle \) and the scalar \( -5 \).
Recall that scalar multiplication of a vector involves multiplying each component of the vector by the scalar.
Multiply the x-component of \( \mathbf{v} \) by \( -5 \): calculate \( -5 \times 4 \).
Multiply the y-component of \( \mathbf{v} \) by \( -5 \): calculate \( -5 \times 3 \).
Combine the results to write the new vector as \( \langle -5 \times 4, -5 \times 3 \rangle \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Representation
Vectors are quantities defined by both magnitude and direction, often represented as ordered pairs or components in coordinate form, such as u = 〈x, y〉. Understanding how vectors are written and interpreted is essential for performing operations like addition, scalar multiplication, and more.
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Introduction to Vectors
Scalar Multiplication of Vectors
Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). This operation changes the vector's magnitude and possibly its direction (if the scalar is negative), but not its direction relative to the scalar's sign. For example, multiplying v = 〈4, 3〉 by -5 results in 〈-20, -15〉.
Recommended video:
05:05Multiplying Vectors By Scalars
Component-wise Operations
Vector operations like addition, subtraction, and scalar multiplication are performed component-wise, meaning each corresponding component is operated on individually. This approach simplifies calculations and helps visualize vector transformations in the coordinate plane.
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04:12Algebraic Operations on Vectors
Related Practice
Textbook Question
Solve each triangle. See Examples 2 and 3.
a = 3.0 ft, b = 5.0 ft, c = 6.0 ft
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Textbook Question
A force of 25 lb is required to hold an 80-lb crate on a hill. What angle does the hill make with the horizontal?
Textbook Question
A force of 28.7 lb makes an angle of 42° 10′ with a second force. The resultant of the two forces makes an angle of 32° 40′ with the first force. Find the magnitudes of the second force and of the resultant.
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Textbook Question
Radio direction finders are placed at points A and B, which are 3.46 mi apart on an east-west line, with A west of B. From A the bearing of a certain radio transmitter is 47.7°, and from B the bearing is 302.5°. Find the distance of the transmitter from A.
Textbook Question
A force of 176 lb makes an angle of 78° 50′ with a second force. The resultant of the two forces makes an angle of 41° 10′ with the first force. Find the magnitudes of the second force and of the resultant.
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