**Educational Website Content:** --- ### Vector Projections Exercise **Problem Statement:** Given two vectors in a 2-dimensional space: \[ \mathbf{v} = 3\mathbf{i} + \mathbf{j} \] \[ \mathbf{w} = -3\mathbf{i} - \mathbf{j} \] Determine the projection of \(\mathbf{v}\) onto \(\mathbf{w}\). **Question:** If \(\mathbf{v} = 3\mathbf{i} + \mathbf{j}\) and \(\mathbf{w} = -3\mathbf{i} - \mathbf{j}\), then the projection of \(\mathbf{v}\) onto \(\mathbf{w}\) is: - \(\mathbf{A.} \ -3\mathbf{i} - \mathbf{j}\) - \(\mathbf{B.} \ 3\mathbf{i} + \mathbf{j}\) - \(\mathbf{C.} \ 2\mathbf{i} + 3\mathbf{j}\) - \(\mathbf{D.} \ -2\mathbf{i} - 3\mathbf{j}\) --- **Solution Explanation:** To find the projection of \(\mathbf{v}\) onto \(\mathbf{w}\), use the formula: \[ \text{proj}_\mathbf{w}\mathbf{v} = \left(\frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}}\right) \mathbf{w} \] Where \(\mathbf{v} \cdot \mathbf{w}\) is the dot product of \(\mathbf{v}\) and \(\mathbf{w}\), and \(\mathbf{w} \cdot \mathbf{w}\) is the dot product of \(\mathbf{w}\) with itself. 1. Calculate the dot product \(\mathbf{v} \cdot \mathbf{w}\): \[ \mathbf{v} \cdot \mathbf{w} = (3)(-3) + (1)(-1) = -9 - 1 = -10 \] 2. Calculate the dot product \(\mathbf{w} \cdot \mathbf{w}\): \[ \mathbf{w} \cdot \mathbf{w} = (-3)^2
**Educational Website Content:** --- ### Vector Projections Exercise **Problem Statement:** Given two vectors in a 2-dimensional space: \[ \mathbf{v} = 3\mathbf{i} + \mathbf{j} \] \[ \mathbf{w} = -3\mathbf{i} - \mathbf{j} \] Determine the projection of \(\mathbf{v}\) onto \(\mathbf{w}\). **Question:** If \(\mathbf{v} = 3\mathbf{i} + \mathbf{j}\) and \(\mathbf{w} = -3\mathbf{i} - \mathbf{j}\), then the projection of \(\mathbf{v}\) onto \(\mathbf{w}\) is: - \(\mathbf{A.} \ -3\mathbf{i} - \mathbf{j}\) - \(\mathbf{B.} \ 3\mathbf{i} + \mathbf{j}\) - \(\mathbf{C.} \ 2\mathbf{i} + 3\mathbf{j}\) - \(\mathbf{D.} \ -2\mathbf{i} - 3\mathbf{j}\) --- **Solution Explanation:** To find the projection of \(\mathbf{v}\) onto \(\mathbf{w}\), use the formula: \[ \text{proj}_\mathbf{w}\mathbf{v} = \left(\frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}}\right) \mathbf{w} \] Where \(\mathbf{v} \cdot \mathbf{w}\) is the dot product of \(\mathbf{v}\) and \(\mathbf{w}\), and \(\mathbf{w} \cdot \mathbf{w}\) is the dot product of \(\mathbf{w}\) with itself. 1. Calculate the dot product \(\mathbf{v} \cdot \mathbf{w}\): \[ \mathbf{v} \cdot \mathbf{w} = (3)(-3) + (1)(-1) = -9 - 1 = -10 \] 2. Calculate the dot product \(\mathbf{w} \cdot \mathbf{w}\): \[ \mathbf{w} \cdot \mathbf{w} = (-3)^2
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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![**Educational Website Content:**
---
### Vector Projections Exercise
**Problem Statement:**
Given two vectors in a 2-dimensional space:
\[ \mathbf{v} = 3\mathbf{i} + \mathbf{j} \]
\[ \mathbf{w} = -3\mathbf{i} - \mathbf{j} \]
Determine the projection of \(\mathbf{v}\) onto \(\mathbf{w}\).
**Question:**
If \(\mathbf{v} = 3\mathbf{i} + \mathbf{j}\) and \(\mathbf{w} = -3\mathbf{i} - \mathbf{j}\), then the projection of \(\mathbf{v}\) onto \(\mathbf{w}\) is:
- \(\mathbf{A.} \ -3\mathbf{i} - \mathbf{j}\)
- \(\mathbf{B.} \ 3\mathbf{i} + \mathbf{j}\)
- \(\mathbf{C.} \ 2\mathbf{i} + 3\mathbf{j}\)
- \(\mathbf{D.} \ -2\mathbf{i} - 3\mathbf{j}\)
---
**Solution Explanation:**
To find the projection of \(\mathbf{v}\) onto \(\mathbf{w}\), use the formula:
\[ \text{proj}_\mathbf{w}\mathbf{v} = \left(\frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}}\right) \mathbf{w} \]
Where \(\mathbf{v} \cdot \mathbf{w}\) is the dot product of \(\mathbf{v}\) and \(\mathbf{w}\), and \(\mathbf{w} \cdot \mathbf{w}\) is the dot product of \(\mathbf{w}\) with itself.
1. Calculate the dot product \(\mathbf{v} \cdot \mathbf{w}\):
\[ \mathbf{v} \cdot \mathbf{w} = (3)(-3) + (1)(-1) = -9 - 1 = -10 \]
2. Calculate the dot product \(\mathbf{w} \cdot \mathbf{w}\):
\[ \mathbf{w} \cdot \mathbf{w} = (-3)^2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc5bcc65-7283-4529-b9b7-ffbb90a5afc1%2F80625fb1-5fc4-4358-a7f2-2686f09e7036%2Fyg4abmrf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Website Content:**
---
### Vector Projections Exercise
**Problem Statement:**
Given two vectors in a 2-dimensional space:
\[ \mathbf{v} = 3\mathbf{i} + \mathbf{j} \]
\[ \mathbf{w} = -3\mathbf{i} - \mathbf{j} \]
Determine the projection of \(\mathbf{v}\) onto \(\mathbf{w}\).
**Question:**
If \(\mathbf{v} = 3\mathbf{i} + \mathbf{j}\) and \(\mathbf{w} = -3\mathbf{i} - \mathbf{j}\), then the projection of \(\mathbf{v}\) onto \(\mathbf{w}\) is:
- \(\mathbf{A.} \ -3\mathbf{i} - \mathbf{j}\)
- \(\mathbf{B.} \ 3\mathbf{i} + \mathbf{j}\)
- \(\mathbf{C.} \ 2\mathbf{i} + 3\mathbf{j}\)
- \(\mathbf{D.} \ -2\mathbf{i} - 3\mathbf{j}\)
---
**Solution Explanation:**
To find the projection of \(\mathbf{v}\) onto \(\mathbf{w}\), use the formula:
\[ \text{proj}_\mathbf{w}\mathbf{v} = \left(\frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}}\right) \mathbf{w} \]
Where \(\mathbf{v} \cdot \mathbf{w}\) is the dot product of \(\mathbf{v}\) and \(\mathbf{w}\), and \(\mathbf{w} \cdot \mathbf{w}\) is the dot product of \(\mathbf{w}\) with itself.
1. Calculate the dot product \(\mathbf{v} \cdot \mathbf{w}\):
\[ \mathbf{v} \cdot \mathbf{w} = (3)(-3) + (1)(-1) = -9 - 1 = -10 \]
2. Calculate the dot product \(\mathbf{w} \cdot \mathbf{w}\):
\[ \mathbf{w} \cdot \mathbf{w} = (-3)^2
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