In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar.
w - v

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

Recall that vector subtraction \( \mathbf{w} - \mathbf{v} \) means subtracting the corresponding components of \( \mathbf{v} \) from \( \mathbf{w} \).

Write the subtraction component-wise: \( (w_x - v_x)\mathbf{i} + (w_y - v_y)\mathbf{j} \), where \( w_x \) and \( w_y \) are the components of \( \mathbf{w} \), and \( v_x \) and \( v_y \) are the components of \( \mathbf{v} \).

Substitute the components: \( (-2 - 1)\mathbf{i} + (7 - (-5))\mathbf{j} \).

Simplify the expressions inside the parentheses to find the resulting vector components.

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주요 개념

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Vector Representation in Component Form

Vectors can be expressed in terms of their components along the standard unit vectors i and j, representing the x and y directions respectively. For example, v = i - 5j means the vector has components (1, -5). This form allows for straightforward algebraic operations on vectors.

Vector Subtraction

Vector subtraction involves subtracting corresponding components of two vectors. Given vectors v = (v_x, v_y) and w = (w_x, w_y), the difference w - v is (w_x - v_x, w_y - v_y). This operation results in a new vector representing the displacement from v to w.

Geometric Interpretation of Vector Operations

Vector subtraction can be visualized as finding the vector pointing from the tip of v to the tip of w. This helps in understanding relative positions and directions in the plane, which is essential in physics and engineering contexts.

In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar.w - v