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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![**Problem Statement:**
Find **u ⋅ v**.
Given that:
\[ \mathbf{u} = 7\mathbf{i} - 2\mathbf{j} \]
\[ \mathbf{v} = \mathbf{i} - \mathbf{j} \]
\[ \mathbf{u} \cdot \mathbf{v} = \boxed{} \]
**Explanation:**
To compute the dot product of the vectors \(\mathbf{u}\) and \(\mathbf{v}\):
1. Express vector \(\mathbf{u}\) as \(\mathbf{u} = 7\mathbf{i} - 2\mathbf{j}\).
2. Express vector \(\mathbf{v}\) as \(\mathbf{v} = \mathbf{i} - \mathbf{j}\).
3. The dot product formula is given by:
\[ \mathbf{u} \cdot \mathbf{v} = u_x v_x + u_y v_y \]
4. Substitute the respective components:
- \(u_x = 7\)
- \(u_y = -2\)
- \(v_x = 1\)
- \(v_y = -1\)
5. Perform the multiplication and addition:
\[ \mathbf{u} \cdot \mathbf{v} = (7)(1) + (-2)(-1) = 7 + 2 = 9 \]
Thus,
\[ \mathbf{u} \cdot \mathbf{v} = 9 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc23443b2-575f-41b5-b981-5d857bb6e2d8%2F65e0914c-8a42-4976-b3be-d6611dd1e9ad%2Foumypv5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find **u ⋅ v**.
Given that:
\[ \mathbf{u} = 7\mathbf{i} - 2\mathbf{j} \]
\[ \mathbf{v} = \mathbf{i} - \mathbf{j} \]
\[ \mathbf{u} \cdot \mathbf{v} = \boxed{} \]
**Explanation:**
To compute the dot product of the vectors \(\mathbf{u}\) and \(\mathbf{v}\):
1. Express vector \(\mathbf{u}\) as \(\mathbf{u} = 7\mathbf{i} - 2\mathbf{j}\).
2. Express vector \(\mathbf{v}\) as \(\mathbf{v} = \mathbf{i} - \mathbf{j}\).
3. The dot product formula is given by:
\[ \mathbf{u} \cdot \mathbf{v} = u_x v_x + u_y v_y \]
4. Substitute the respective components:
- \(u_x = 7\)
- \(u_y = -2\)
- \(v_x = 1\)
- \(v_y = -1\)
5. Perform the multiplication and addition:
\[ \mathbf{u} \cdot \mathbf{v} = (7)(1) + (-2)(-1) = 7 + 2 = 9 \]
Thus,
\[ \mathbf{u} \cdot \mathbf{v} = 9 \]
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