Textbook Question
In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given. ||v|| = 12, θ = 225°
Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 4, Problem 53
In Exercises 53–56, letu = -2i + 3j, v = 6i - j, w = -3i.Find each specified vector or scalar.4u - (2v - w)
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Identify the given vectors: \( \mathbf{u} = -2\mathbf{i} + 3\mathbf{j} \), \( \mathbf{v} = 6\mathbf{i} - \mathbf{j} \), and \( \mathbf{w} = -3\mathbf{i} \).
Calculate \( 2\mathbf{v} \) by multiplying each component of \( \mathbf{v} \) by 2: \( 2\mathbf{v} = 2(6\mathbf{i} - \mathbf{j}) = 12\mathbf{i} - 2\mathbf{j} \).
Subtract \( \mathbf{w} \) from \( 2\mathbf{v} \): \( 2\mathbf{v} - \mathbf{w} = (12\mathbf{i} - 2\mathbf{j}) - (-3\mathbf{i}) = 12\mathbf{i} - 2\mathbf{j} + 3\mathbf{i} = 15\mathbf{i} - 2\mathbf{j} \).
Calculate \( 4\mathbf{u} \) by multiplying each component of \( \mathbf{u} \) by 4: \( 4\mathbf{u} = 4(-2\mathbf{i} + 3\mathbf{j}) = -8\mathbf{i} + 12\mathbf{j} \).
Subtract \( (2\mathbf{v} - \mathbf{w}) \) from \( 4\mathbf{u} \): \( 4\mathbf{u} - (2\mathbf{v} - \mathbf{w}) = (-8\mathbf{i} + 12\mathbf{j}) - (15\mathbf{i} - 2\mathbf{j}) = -8\mathbf{i} + 12\mathbf{j} - 15\mathbf{i} + 2\mathbf{j} = -23\mathbf{i} + 14\mathbf{j} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Operations
Vector operations include addition, subtraction, and scalar multiplication. In this context, vectors are treated as quantities with both magnitude and direction, represented in component form. Understanding how to manipulate vectors through these operations is essential for solving problems involving multiple vectors.
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Algebraic Operations on Vectors
Scalar Multiplication
Scalar multiplication involves multiplying a vector by a scalar (a real number), which scales the vector's magnitude without changing its direction. For example, multiplying the vector u = -2i + 3j by a scalar 4 results in the vector -8i + 12j. This concept is crucial for transforming vectors in the given expression.
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Vector Subtraction
Vector subtraction is the process of finding the difference between two vectors, which can be visualized as adding the negative of one vector to another. For instance, subtracting vector w from 2v involves changing the direction of w and then adding it to 2v. This operation is key to simplifying the expression in the problem.
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Related Practice
Textbook Question
Find the magnitude ||v||, to the nearest hundredth, and the direction angle θ, to the nearest tenth of a degree, for each given vector v.
v = 2i - 8j
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Textbook Question
In Exercises 61–64, find the magnitude ||v||, to the nearest hundredth, and the direction angle θ, to the nearest tenth of a degree, for each given vector v. v = -10i + 15j
Textbook Question
Write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given.
||v|| = 10, θ = 330°
Textbook Question
In Exercises 53–56, let u = -2i + 3j, v = 6i - j, w = -3i. Find each specified vector or scalar. ||u + v||² - ||u - v||²
10
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Textbook Question
In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given. ||v|| = 1/2, θ = 113°