In Exercises 42–43, find projᵥᵥv. Then decompose v into two vectors, v₁ and v₂ where v₁ is parallel to w and v₂ is orthogonal to w.
v = -2i + 5j, w = 5i + 4j

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Identify the vectors \( \mathbf{v} = -2\mathbf{i} + 5\mathbf{j} \) and \( \mathbf{w} = 5\mathbf{i} + 4\mathbf{j} \).

Recall the formula for the projection of \( \mathbf{v} \) onto \( \mathbf{w} \): \[ \text{proj}_{\mathbf{w}} \mathbf{v} = \left( \frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} \right) \mathbf{w} \] where \( \mathbf{v} \cdot \mathbf{w} \) is the dot product of \( \mathbf{v} \) and \( \mathbf{w} \).

Calculate the dot product \( \mathbf{v} \cdot \mathbf{w} = (-2)(5) + (5)(4) \).

Calculate the dot product \( \mathbf{w} \cdot \mathbf{w} = (5)(5) + (4)(4) \).

Use the values from the dot products to find \( \text{proj}_{\mathbf{w}} \mathbf{v} \) by multiplying the scalar \( \frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} \) by the vector \( \mathbf{w} \). Then, decompose \( \mathbf{v} \) into: - \( \mathbf{v}_1 = \text{proj}_{\mathbf{w}} \mathbf{v} \) (parallel to \( \mathbf{w} \)) - \( \mathbf{v}_2 = \mathbf{v} - \mathbf{v}_1 \) (orthogonal to \( \mathbf{w} \))

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주요 개념

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Vector Projection

Vector projection of v onto w, denoted proj_w v, is the component of v that points in the direction of w. It is calculated using the formula proj_w v = (v · w / ||w||²) w, where '·' is the dot product and ||w|| is the magnitude of w. This concept helps in breaking down vectors into parallel components.

Dot Product of Vectors

The dot product of two vectors v and w is a scalar defined as v · w = v₁w₁ + v₂w₂ for 2D vectors. It measures how much one vector extends in the direction of another and is essential for finding projections and angles between vectors.

Vector Decomposition into Parallel and Orthogonal Components

Any vector v can be decomposed into two vectors: v₁ parallel to w and v₂ orthogonal to w, such that v = v₁ + v₂. Here, v₁ = proj_w v, and v₂ = v - v₁. This decomposition is useful in many applications like resolving forces or analyzing vector components.

In Exercises 42–43, find projᵥᵥv. Then decompose v into two vectors, v₁ and v₂ where v₁ is parallel to w and v₂ is orthogonal to w.v = -2i + 5j, w = 5i + 4j