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

검증된 단계별 안내

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

Recall that vector subtraction \( \mathbf{v} - \mathbf{u} \) is performed by subtracting the corresponding components of \( \mathbf{u} \) from \( \mathbf{v} \).

Subtract the \( \mathbf{i} \)-components: \( -3 - 2 = -5 \).

Subtract the \( \mathbf{j} \)-components: \( 7 - (-5) = 7 + 5 = 12 \).

Write the resulting vector as \( \mathbf{v} - \mathbf{u} = -5\mathbf{i} + 12\mathbf{j} \).

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주요 개념

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Vector Representation in Component Form

Vectors in two dimensions can be expressed as 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. Understanding this form allows for straightforward vector operations like addition and subtraction.

Vector Subtraction

Vector subtraction involves subtracting corresponding components of two vectors. For vectors v and u, v - u is found by subtracting the x-components and y-components separately, resulting in a new vector. This operation is essential for finding the difference or relative position between vectors.

Unit Vectors i and j

Unit vectors i and j represent the standard basis vectors along the x-axis and y-axis, respectively. They have a magnitude of one and direction along their axes. Expressing vectors in terms of i and j simplifies calculations and visualization in the plane.

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