Textbook Question
A ship leaves port on a bearing of 34.0° and travels 10.4 mi. The ship then turns due east and travels 4.6 mi. How far is the ship from port, and what is its bearing from port?
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 43c
Given vectors u and v, find: v - 3u.
u = 〈-1, 2〉, v = 〈3, 0〉
Verified step by step guidance1
Identify the given vectors: \( \mathbf{u} = \langle -1, 2 \rangle \) and \( \mathbf{v} = \langle 3, 0 \rangle \).
Understand that the expression \( \mathbf{v} - 3\mathbf{u} \) means you need to multiply vector \( \mathbf{u} \) by the scalar 3, then subtract the resulting vector from \( \mathbf{v} \).
Calculate the scalar multiplication: multiply each component of \( \mathbf{u} \) by 3, which gives \( 3 \mathbf{u} = \langle 3 \times (-1), 3 \times 2 \rangle = \langle -3, 6 \rangle \).
Perform the vector subtraction by subtracting the corresponding components of \( 3\mathbf{u} \) from \( \mathbf{v} \): \( \mathbf{v} - 3\mathbf{u} = \langle 3 - (-3), 0 - 6 \rangle \).
Simplify the subtraction inside the components to get the resulting vector.
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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 tuples in coordinate form, such as u = 〈x, y〉. Understanding how to interpret these components is essential for performing operations like addition, subtraction, and scalar multiplication.
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Introduction to Vectors
Scalar Multiplication of Vectors
Scalar multiplication involves multiplying each component of a vector by a real number (scalar). For example, multiplying vector u = 〈x, y〉 by scalar 3 results in 〈3x, 3y〉. This operation changes the vector's magnitude but not its direction unless the scalar is negative.
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05:05Multiplying Vectors By Scalars
Vector Addition and Subtraction
Adding or subtracting vectors is done component-wise: for vectors a = 〈a1, a2〉 and b = 〈b1, b2〉, a ± b = 〈a1 ± b1, a2 ± b2〉. This principle allows combining vectors or finding the difference between them, which is crucial for solving expressions like v - 3u.
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05:29Adding Vectors Geometrically
Related Practice
Textbook Question
Two people are carrying a box. One person exerts a force of 150 lb at an angle of 62.4° with the horizontal. The other person exerts a force of 114 lb at an angle of 54.9°. Find the weight of the box.
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Textbook Question
Given vectors u and v, find: 2u + 3v.
u = 〈-1, 2〉, v = 〈3, 0〉
Textbook Question
Write each vector in the form a i + b j.
〈6, -3〉
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Textbook Question
Given vectors u and v, find: 2u.
u = 〈-1, 2〉, v = 〈3, 0〉
Textbook Question
A crate is supported by two ropes. One rope makes an angle of 46° 20′ with the horizontal and has a tension of 89.6 lb on it. The other rope is horizontal. Find the weight of the crate and the tension in the horizontal rope.