Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Vector Decomposition and Projection**
Decompose **a** = ⟨4, -1, 0⟩ into parts parallel and perpendicular to **b** = ⟨0, 1, 1⟩. What is the projection of **a** onto **b**?
When decomposing a vector **a** into components parallel and perpendicular to another vector **b**, the following steps are taken:
1. **Projection of **a** onto **b****:
- The projection of **a** onto **b** is given by the formula:
\[
\text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \right) \mathbf{b}
\]
2. **Calculation of the Projection**:
- Compute the dot product: **a** ⋅ **b** = 4(0) + (-1)(1) + 0(1) = -1
- Compute **b** ⋅ **b** = (0)^2 + (1)^2 + (1)^2 = 2
- Thus, the projection of **a** onto **b**:
\[
\text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{-1}{2} \right) \mathbf{b} = \left( \frac{-1}{2} \right) \langle0, 1, 1\rangle = \langle 0, -\frac{1}{2}, -\frac{1}{2} \rangle
\]
3. **Component of **a** Perpendicular to **b****:
- The perpendicular component is found using the equation:
\[
\mathbf{a}_{\perp} = \mathbf{a} - \text{proj}_{\mathbf{b}} \mathbf{a}
\]
- Calculate:
\[
\mathbf{a}_{\perp} = \langle 4, -1, 0 \rangle - \langle 0, -\frac{1}{2}, -\frac{1}{2} \rangle = \langle 4, -\frac{1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec99e2ce-f146-4dfd-8358-2828b726596b%2Ff2a66622-b190-43e4-b246-64078a6e0cf3%2F326aj8e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Vector Decomposition and Projection**
Decompose **a** = ⟨4, -1, 0⟩ into parts parallel and perpendicular to **b** = ⟨0, 1, 1⟩. What is the projection of **a** onto **b**?
When decomposing a vector **a** into components parallel and perpendicular to another vector **b**, the following steps are taken:
1. **Projection of **a** onto **b****:
- The projection of **a** onto **b** is given by the formula:
\[
\text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \right) \mathbf{b}
\]
2. **Calculation of the Projection**:
- Compute the dot product: **a** ⋅ **b** = 4(0) + (-1)(1) + 0(1) = -1
- Compute **b** ⋅ **b** = (0)^2 + (1)^2 + (1)^2 = 2
- Thus, the projection of **a** onto **b**:
\[
\text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{-1}{2} \right) \mathbf{b} = \left( \frac{-1}{2} \right) \langle0, 1, 1\rangle = \langle 0, -\frac{1}{2}, -\frac{1}{2} \rangle
\]
3. **Component of **a** Perpendicular to **b****:
- The perpendicular component is found using the equation:
\[
\mathbf{a}_{\perp} = \mathbf{a} - \text{proj}_{\mathbf{b}} \mathbf{a}
\]
- Calculate:
\[
\mathbf{a}_{\perp} = \langle 4, -1, 0 \rangle - \langle 0, -\frac{1}{2}, -\frac{1}{2} \rangle = \langle 4, -\frac{1
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