In Exercises 33–38, find projᵥᵥ v. Then decompose v into two vectors, v₁ and v₂, where v₁ is parallel to w and v₂ is orthogonal to w. v = 3i - 2j, w = i - j

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Identify the vectors \( \mathbf{v} = 3\mathbf{i} - 2\mathbf{j} \) and \( \mathbf{w} = \mathbf{i} - \mathbf{j} \).

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 \( \cdot \) denotes the dot product.

Calculate the dot product \( \mathbf{v} \cdot \mathbf{w} \) by multiplying corresponding components and summing: \[ \mathbf{v} \cdot \mathbf{w} = (3)(1) + (-2)(-1) \]

Calculate the dot product \( \mathbf{w} \cdot \mathbf{w} \) similarly: \[ \mathbf{w} \cdot \mathbf{w} = (1)(1) + (-1)(-1) \]

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, find \( \mathbf{v}_1 = \text{proj}_{\mathbf{w}} \mathbf{v} \) (parallel component) and \( \mathbf{v}_2 = \mathbf{v} - \mathbf{v}_1 \) (orthogonal component).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 · w) w, where '·' denotes the dot product. This concept helps in finding how much of one vector lies along another.

Dot Product of Vectors

The dot product is an algebraic operation that takes two vectors and returns a scalar. It is computed as v · w = v₁w₁ + v₂w₂ for 2D vectors. The dot product is essential for finding projections and determining angles between vectors.

Vector Decomposition into Parallel and Orthogonal Components

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