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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Question
![**Find \(\mathbf{u} \cdot \mathbf{v}\), \(\mathbf{u} \cdot \mathbf{u}\), \(\|\mathbf{v}\|^2\), \((\mathbf{u} \cdot \mathbf{v})\mathbf{v}\), and \(\mathbf{u} \cdot (3\mathbf{v})\).**
Vectors:
- \(\mathbf{u} = \langle 2, 11 \rangle\)
- \(\mathbf{v} = \langle -5, 2 \rangle\)
Calculations:
- **(a) \(\mathbf{u} \cdot \mathbf{v}\)**
\[
(2)(-5) + (11)(2) = -10 + 22 = 12
\]
- **(b) \(\mathbf{u} \cdot \mathbf{u}\)**
\[
(2)(2) + (11)(11) = 4 + 121 = 125
\]
- **(c) \(\|\mathbf{v}\|^2\)**
\[
(-5)^2 + (2)^2 = 25 + 4 = 29
\]
- **(d) \((\mathbf{u} \cdot \mathbf{v})\mathbf{v}\)**
\[
12 \cdot \langle -5, 2 \rangle = \langle 12(-5), 12(2) \rangle = \langle -60, 24 \rangle
\]
- **(e) \(\mathbf{u} \cdot (3\mathbf{v})\)**
\[
\mathbf{u} \cdot \langle 3(-5), 3(2) \rangle = \langle -15, 6 \rangle
\]
\[
(2)(-15) + (11)(6) = -30 + 66 = 36
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba13667f-f60c-45eb-87f2-7662478f2e6d%2F2d15261a-b72f-4af3-8a0d-55c32216a03e%2Fhqrwp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Find \(\mathbf{u} \cdot \mathbf{v}\), \(\mathbf{u} \cdot \mathbf{u}\), \(\|\mathbf{v}\|^2\), \((\mathbf{u} \cdot \mathbf{v})\mathbf{v}\), and \(\mathbf{u} \cdot (3\mathbf{v})\).**
Vectors:
- \(\mathbf{u} = \langle 2, 11 \rangle\)
- \(\mathbf{v} = \langle -5, 2 \rangle\)
Calculations:
- **(a) \(\mathbf{u} \cdot \mathbf{v}\)**
\[
(2)(-5) + (11)(2) = -10 + 22 = 12
\]
- **(b) \(\mathbf{u} \cdot \mathbf{u}\)**
\[
(2)(2) + (11)(11) = 4 + 121 = 125
\]
- **(c) \(\|\mathbf{v}\|^2\)**
\[
(-5)^2 + (2)^2 = 25 + 4 = 29
\]
- **(d) \((\mathbf{u} \cdot \mathbf{v})\mathbf{v}\)**
\[
12 \cdot \langle -5, 2 \rangle = \langle 12(-5), 12(2) \rangle = \langle -60, 24 \rangle
\]
- **(e) \(\mathbf{u} \cdot (3\mathbf{v})\)**
\[
\mathbf{u} \cdot \langle 3(-5), 3(2) \rangle = \langle -15, 6 \rangle
\]
\[
(2)(-15) + (11)(6) = -30 + 66 = 36
\]
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