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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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 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.

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.

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.