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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 4 - Laws of Sines and Cosines; VectorsProblem 5
Chapter 4, Problem 5

In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. 3v - 4w

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Identify the given vectors: \( \mathbf{v} = -5\mathbf{i} + 2\mathbf{j} \) and \( \mathbf{w} = 2\mathbf{i} - 4\mathbf{j} \).
Calculate the scalar multiplication of vector \( \mathbf{v} \) by 3: multiply each component of \( \mathbf{v} \) by 3, resulting in \( 3\mathbf{v} = 3(-5\mathbf{i}) + 3(2\mathbf{j}) \).
Calculate the scalar multiplication of vector \( \mathbf{w} \) by 4: multiply each component of \( \mathbf{w} \) by 4, resulting in \( 4\mathbf{w} = 4(2\mathbf{i}) + 4(-4\mathbf{j}) \).
Form the expression \( 3\mathbf{v} - 4\mathbf{w} \) by subtracting the components of \( 4\mathbf{w} \) from the corresponding components of \( 3\mathbf{v} \).
Combine the resulting components to write the final vector in the form \( a\mathbf{i} + b\mathbf{j} \), where \( a \) and \( b \) are the calculated values.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vector Addition and Scalar Multiplication

Vector addition involves adding corresponding components of vectors, while scalar multiplication scales each component by a given number. For example, multiplying vector v by 3 means multiplying each component of v by 3. These operations are fundamental for combining and scaling vectors.
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Multiplying Vectors By Scalars

Component Form of Vectors

Vectors in two dimensions can be expressed in component form as v = ai + bj, where a and b are the components along the x and y axes, respectively. Understanding this form allows for straightforward arithmetic operations on vectors by handling their components separately.
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Linear Combination of Vectors

A linear combination involves multiplying vectors by scalars and then adding the results. In this problem, 3v - 4w is a linear combination, meaning you multiply v by 3, w by 4, and subtract the results component-wise to find the resulting vector.
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Related Practice
Textbook Question

In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. C = 50°, a = 3, c = 1

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Textbook Question

In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = -6i - 5j, w = -10i - 8j

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Textbook Question

In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

Textbook Question
In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. c. Find the exact value of the expression.cos 50° cos 20° + sin 50° sin 20°
Textbook Question

In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. v ⋅ w

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Textbook Question

In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

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views