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

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.

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