In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
3w + 2v

검증된 단계별 안내

Identify the given vectors: \( \mathbf{u} = 2\mathbf{i} - 5\mathbf{j} \), \( \mathbf{v} = -3\mathbf{i} + 7\mathbf{j} \), and \( \mathbf{w} = -\mathbf{i} - 6\mathbf{j} \).

Calculate the scalar multiplication of vector \( \mathbf{w} \) by 3: multiply each component of \( \mathbf{w} \) by 3, resulting in \( 3\mathbf{w} = 3(-\mathbf{i}) + 3(-6\mathbf{j}) \).

Calculate the scalar multiplication of vector \( \mathbf{v} \) by 2: multiply each component of \( \mathbf{v} \) by 2, resulting in \( 2\mathbf{v} = 2(-3\mathbf{i}) + 2(7\mathbf{j}) \).

Add the resulting vectors from the previous two steps component-wise: add the \( \mathbf{i} \) components together and the \( \mathbf{j} \) components together to find \( 3\mathbf{w} + 2\mathbf{v} \).

Write the final vector in the form \( a\mathbf{i} + b\mathbf{j} \), where \( a \) and \( b \) are the sums of the respective components.

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주요 개념

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

Vector Representation in Component Form

Vectors in two dimensions can be expressed as a combination of unit vectors i and j, representing the x and y components respectively. For example, u = 2i - 5j means the vector has an x-component of 2 and a y-component of -5. Understanding this form allows for straightforward vector addition and scalar multiplication.

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). For instance, multiplying vector w by 3 means multiplying both its i and j components by 3, resulting in a new vector scaled in magnitude but with the same direction if the scalar is positive.

Vector Addition

Vector addition is performed by adding corresponding components of two vectors. For example, adding 2v and 3w requires adding the x-components of 2v and 3w together, and similarly for the y-components, resulting in a new vector that combines both directions and magnitudes.

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.3w + 2v