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 = i + 3j, w = -2i + 5j

Verified step by step guidance

1. Calculate the dot product of \( \mathbf{v} \) and \( \mathbf{w} \): \( \mathbf{v} \cdot \mathbf{w} = (1)(-2) + (3)(5) \).
2. Calculate the magnitude squared of \( \mathbf{w} \): \( ||\mathbf{w}||^2 = (-2)^2 + (5)^2 \).
3. Find the projection of \( \mathbf{v} \) onto \( \mathbf{w} \) using the formula: \( \text{proj}_{\mathbf{w}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{w}}{||\mathbf{w}||^2} \mathbf{w} \).
4. Decompose \( \mathbf{v} \) into \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) where \( \mathbf{v}_1 = \text{proj}_{\mathbf{w}} \mathbf{v} \) and \( \mathbf{v}_2 = \mathbf{v} - \mathbf{v}_1 \).
5. Verify that \( \mathbf{v}_2 \) is orthogonal to \( \mathbf{w} \) by checking that \( \mathbf{v}_2 \cdot \mathbf{w} = 0 \).

Key Concepts

Vector Projection

Vector projection is the process of projecting one vector onto another. The projection of vector v onto vector w, denoted as projᵥᵥ w, is calculated using the formula projᵥᵥ w = (v · w / ||w||²) * w, where '·' represents the dot product and ||w|| is the magnitude of vector w. This concept is essential for decomposing vectors into components that are parallel and orthogonal to a given vector.

Dot Product

The dot product is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It is calculated as v · w = v₁w₁ + v₂w₂, where v₁ and v₂ are the components of vector v, and w₁ and w₂ are the components of vector w. The dot product is crucial for finding the angle between vectors and is used in the projection formula.

Orthogonal Vectors

Orthogonal vectors are vectors that are perpendicular to each other, meaning their dot product equals zero. In the context of vector decomposition, once the parallel component (v₁) is found, the orthogonal component (v₂) can be determined by subtracting v₁ from the original vector v. Understanding orthogonality is key to effectively breaking down vectors into their respective components.