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Ch. 7 - Applications of Trigonometry and Vectors
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 7 - Applications of Trigonometry and VectorsProblema 38
Capitolo 8, Problema 38

Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
- 2u + 4v

Guida verificata passo dopo passo
1
Identify the given vectors: \( \mathbf{u} = \langle -2, 5 \rangle \) and \( \mathbf{v} = \langle 4, 3 \rangle \).
Understand that scalar multiplication means multiplying each component of the vector by the scalar. For example, \( 2\mathbf{u} = \langle 2 \times (-2), 2 \times 5 \rangle \).
Calculate \( 2\mathbf{u} \) by multiplying each component of \( \mathbf{u} \) by 2: \( 2\mathbf{u} = \langle -4, 10 \rangle \).
Calculate \( 4\mathbf{v} \) by multiplying each component of \( \mathbf{v} \) by 4: \( 4\mathbf{v} = \langle 16, 12 \rangle \).
Add the resulting vectors component-wise: \( 2\mathbf{u} + 4\mathbf{v} = \langle -4 + 16, 10 + 12 \rangle \).

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Vector Addition and Scalar Multiplication

Vector addition involves adding corresponding components of two vectors to form a new vector. Scalar multiplication means multiplying each component of a vector by a scalar (a real number). These operations allow combining and scaling vectors, essential for expressions like 2u + 4v.
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Multiplying Vectors By Scalars

Component-wise Operations

Vectors in two dimensions are represented by ordered pairs. Operations such as addition and scalar multiplication are performed component-wise, meaning each x-component and y-component is handled separately. This simplifies calculations and helps visualize vector results.
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Algebraic Operations on Vectors

Notation and Vector Representation

Vectors are often denoted by angle brackets, e.g., 〈x, y〉, representing their components along the x and y axes. Understanding this notation is crucial for interpreting and manipulating vectors in problems involving vector arithmetic.
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i & j Notation
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