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

검증된 단계별 안내

Identify the given vector \( w = -\mathbf{i} - 6\mathbf{j} \), which can be written in component form as \( w = (-1, -6) \).

Understand that multiplying a vector by a scalar means multiplying each component of the vector by that scalar.

Set up the scalar multiplication: \( -4w = -4 \times (-1, -6) \).

Multiply each component of \( w \) by \( -4 \): \( -4 \times (-1) \) for the \( \mathbf{i} \) component and \( -4 \times (-6) \) for the \( \mathbf{j} \) component.

Write the resulting vector after multiplication as \( -4w = (4, 24) \), which corresponds to \( 4\mathbf{i} + 24\mathbf{j} \).

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주요 개념

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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 operations like addition, subtraction, and scalar multiplication.

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). This operation changes the magnitude of the vector without altering its direction if the scalar is positive, or reverses the direction if the scalar is negative. For instance, multiplying vector w by -4 scales and reverses w.

Vector Notation and Operations

Vectors are often denoted using unit vectors i and j for clarity in two dimensions. Operations like addition, subtraction, and scalar multiplication are performed component-wise. Recognizing how to manipulate vectors in this notation is essential for solving problems involving vector quantities.

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