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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![### Vectors and Projections
#### Problem Statement
**Objective**: Find the scalar and vector projections of **b** on **a**.
Given vectors:
\[
\mathbf{a} = 2\mathbf{i} + \mathbf{j} + 4\mathbf{k}
\]
\[
\mathbf{b} = \mathbf{j} + \frac{1}{2}\mathbf{k}
\]
#### Step-by-Step Solution
1. **Calculate the Dot Product**: The dot product of vectors \(\mathbf{a}\) and \(\mathbf{b}\) is given by:
\[
\mathbf{a} \cdot \mathbf{b} = (2\mathbf{i} + \mathbf{j} + 4\mathbf{k}) \cdot (\mathbf{j} + \frac{1}{2}\mathbf{k})
\]
Expanding this:
\[
\mathbf{a} \cdot \mathbf{b} = (2 \times 0) + (1 \times 1) + (4 \times \frac{1}{2}) = 0 + 1 + 2 = 3
\]
2. **Calculate the Magnitude of \(\mathbf{a}\)**: The magnitude is given by:
\[
\|\mathbf{a}\| = \sqrt{(2^2 + 1^2 + 4^2)} = \sqrt{4 + 1 + 16} = \sqrt{21}
\]
3. **Scalar Projection**: The scalar projection of \(\mathbf{b}\) on \(\mathbf{a}\) is:
\[
\text{comp}_{\mathbf{a}} \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|} = \frac{3}{\sqrt{21}} = \frac{3}{\sqrt{21}} \times \frac{\sqrt{21}}{\sqrt{21}} = \frac{3\sqrt{21}}{21} = \frac{\sqrt{21}}{7}
\]
4. **Vector Projection**: The vector projection of \(\mathbf{b}\) on \(\mathbf{a}\) is:
\[
\text{proj}_{\mathbf{a}} \mathbf{b} = \left( \frac{\mathbf{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7bb6f29c-bae7-4b63-bf8e-21de0c88ea98%2F09b2438a-5c3b-4aad-8c47-d8a2e52ab1e7%2Fdkgxwkh.jpeg&w=3840&q=75)
Transcribed Image Text:### Vectors and Projections
#### Problem Statement
**Objective**: Find the scalar and vector projections of **b** on **a**.
Given vectors:
\[
\mathbf{a} = 2\mathbf{i} + \mathbf{j} + 4\mathbf{k}
\]
\[
\mathbf{b} = \mathbf{j} + \frac{1}{2}\mathbf{k}
\]
#### Step-by-Step Solution
1. **Calculate the Dot Product**: The dot product of vectors \(\mathbf{a}\) and \(\mathbf{b}\) is given by:
\[
\mathbf{a} \cdot \mathbf{b} = (2\mathbf{i} + \mathbf{j} + 4\mathbf{k}) \cdot (\mathbf{j} + \frac{1}{2}\mathbf{k})
\]
Expanding this:
\[
\mathbf{a} \cdot \mathbf{b} = (2 \times 0) + (1 \times 1) + (4 \times \frac{1}{2}) = 0 + 1 + 2 = 3
\]
2. **Calculate the Magnitude of \(\mathbf{a}\)**: The magnitude is given by:
\[
\|\mathbf{a}\| = \sqrt{(2^2 + 1^2 + 4^2)} = \sqrt{4 + 1 + 16} = \sqrt{21}
\]
3. **Scalar Projection**: The scalar projection of \(\mathbf{b}\) on \(\mathbf{a}\) is:
\[
\text{comp}_{\mathbf{a}} \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|} = \frac{3}{\sqrt{21}} = \frac{3}{\sqrt{21}} \times \frac{\sqrt{21}}{\sqrt{21}} = \frac{3\sqrt{21}}{21} = \frac{\sqrt{21}}{7}
\]
4. **Vector Projection**: The vector projection of \(\mathbf{b}\) on \(\mathbf{a}\) is:
\[
\text{proj}_{\mathbf{a}} \mathbf{b} = \left( \frac{\mathbf{
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