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.
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 46
Write each vector in the form a i + b j.
〈6, -3〉
Verified step by step guidance1
Understand that the vector given in angle bracket notation 〈x, y〉 can be expressed in terms of unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), where \( \mathbf{i} \) is the unit vector in the x-direction and \( \mathbf{j} \) is the unit vector in the y-direction.
Identify the components of the vector: here, the x-component is 6 and the y-component is -3.
Write the vector as a linear combination of the unit vectors: multiply the x-component by \( \mathbf{i} \) and the y-component by \( \mathbf{j} \).
Express the vector as \( 6\mathbf{i} + (-3)\mathbf{j} \).
Simplify the expression by writing it as \( 6\mathbf{i} - 3\mathbf{j} \), which is the vector in the form \( a\mathbf{i} + b\mathbf{j} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Representation in Component Form
A vector in two dimensions can be expressed as a combination of its horizontal and vertical components. These components correspond to the vector's projections along the x-axis and y-axis, respectively, and are typically written as a multiple of unit vectors i (x-direction) and j (y-direction).
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Position Vectors & Component Form
Unit Vectors i and j
Unit vectors i and j are standard basis vectors in the plane, where i represents a vector of length one in the positive x-direction, and j represents a vector of length one in the positive y-direction. Any 2D vector can be written as a linear combination of i and j.
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Converting Coordinate Notation to Vector Form
Given a vector in coordinate form 〈x, y〉, it can be rewritten as x i + y j by multiplying the x-component by i and the y-component by j. This form clearly shows the vector's direction and magnitude along each axis.
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Related Practice
Textbook Question
Given vectors u and v, find: v - 3u.
u = 〈-1, 2〉, v = 〈3, 0〉
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?
Textbook Question
A luxury liner leaves port on a bearing of 110.0° and travels 8.8 mi. It then turns due west and travels 2.4 mi. How far is the liner from port, and what is its bearing from port?
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Textbook Question
Write each vector in the form a i + b j.
〈2, 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.