Textbook Question
Given vectors u and v, find: v - 3u.
u = 〈-1, 2〉, v = 〈3, 0〉
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 43a
Given vectors u and v, find: 2u.
u = 〈-1, 2〉, v = 〈3, 0〉
Verified step by step guidance1
Identify the vector \( \mathbf{u} = \langle -1, 2 \rangle \) and the scalar multiplier, which is 2 in this case.
Recall that multiplying a vector by a scalar means multiplying each component of the vector by that scalar.
Apply the scalar multiplication to vector \( \mathbf{u} \): multiply the first component by 2 and the second component by 2.
Write the resulting vector as \( 2\mathbf{u} = \langle 2 \times (-1), 2 \times 2 \rangle \).
Simplify the components to express the final vector in the form \( \langle a, b \rangle \), where \( a \) and \( b \) are the products from the previous step.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Scalar Multiplication
Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). For example, multiplying vector u = 〈x, y〉 by 2 results in 2u = 〈2x, 2y〉. This operation scales the vector's magnitude without changing its direction.
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Multiplying Vectors By Scalars
Vector Notation and Components
Vectors in two dimensions are represented as ordered pairs 〈x, y〉, where x and y are components along the horizontal and vertical axes. Understanding this notation is essential for performing operations like addition, subtraction, and scalar multiplication.
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06:01i & j Notation
Basic Vector Operations
Basic vector operations include addition, subtraction, and scalar multiplication. These operations follow component-wise rules, allowing manipulation of vectors algebraically. Mastery of these operations is fundamental for solving vector-related problems.
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04:12Algebraic Operations on Vectors
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
Given vectors u and v, find: v - 3u.
u = 2i, v = i + j
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.
Textbook Question
A force of 30.0 lb is required to hold an 80.0-lb pressure washer on an incline. What angle does the incline make with the horizontal?