In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
3v - 4w

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

Multiply vector \( \mathbf{v} \) by the scalar 3: calculate \( 3\mathbf{v} = 3(-3\mathbf{i} + 7\mathbf{j}) \).

Multiply vector \( \mathbf{w} \) by the scalar 4: calculate \( 4\mathbf{w} = 4(-\mathbf{i} - 6\mathbf{j}) \).

Subtract the vector \( 4\mathbf{w} \) from \( 3\mathbf{v} \): compute \( 3\mathbf{v} - 4\mathbf{w} \) by subtracting corresponding components.

Write the resulting vector in component form \( a\mathbf{i} + b\mathbf{j} \) by combining the \( \mathbf{i} \) and \( \mathbf{j} \) components from the subtraction.

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주요 개념

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Vector Representation in Component Form

Vectors in two dimensions can be expressed as components along the i (x-axis) and j (y-axis) unit vectors. For example, u = 2i - 5j means the vector has an x-component of 2 and a y-component of -5. This form allows for straightforward addition, subtraction, and scalar multiplication of vectors.

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). For instance, multiplying vector v 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 and Subtraction

Adding or subtracting vectors is done component-wise by adding or subtracting their corresponding i and j components. For example, to find 3v - 4w, first multiply each vector by its scalar, then subtract the resulting components to get the final vector.

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.3v - 4w