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
![**Vectors and Angles Between Them**
*Graph Explanation:*
This graph is a coordinate plane marked with x and y-axes, each ranging from -6 to 6 with grid intervals of 1 unit. Two vectors, labeled \( \mathbf{u} \) and \( \mathbf{v} \), originate from the origin (0,0). The first vector, \( \mathbf{u} \), points to the coordinates (1,3), indicating it has a component of 1 in the x-direction and 3 in the y-direction. The second vector, \( \mathbf{v} \), points to the coordinates (2,0), indicating it has a component of 2 in the x-direction and 0 in the y-direction.
*Mathematical Explanation:*
The accompanying mathematical problem requires setting the vectors \( \mathbf{u} \) and \( \mathbf{v} \) such that the equation involving the dot product and magnitudes of the vectors holds true:
\[ \cos^{-1} \left( \frac{\mathbf{v} \cdot \mathbf{u}}{|\mathbf{v}| |\mathbf{u}|} \right) = \frac{\pi}{3} \]
In this case, \( \mathbf{v} \cdot \mathbf{u} \) represents the dot product of vectors \( \mathbf{v} \) and \( \mathbf{u} \), and \( |\mathbf{v}| \) and \( |\mathbf{u}| \) represent the magnitudes (lengths) of vectors \( \mathbf{v} \) and \( \mathbf{u} \) respectively.
The equation states that the angle between vectors \( \mathbf{v} \) and \( \mathbf{u} \) should be \( \frac{\pi}{3} \) radians, equivalent to 60 degrees.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25b167f9-3118-479d-a9fb-f8c446c41f1a%2F22861537-17d8-4006-9e8c-45250e36bb5b%2F1ze8q98_processed.png&w=3840&q=75)
Transcribed Image Text:**Vectors and Angles Between Them**
*Graph Explanation:*
This graph is a coordinate plane marked with x and y-axes, each ranging from -6 to 6 with grid intervals of 1 unit. Two vectors, labeled \( \mathbf{u} \) and \( \mathbf{v} \), originate from the origin (0,0). The first vector, \( \mathbf{u} \), points to the coordinates (1,3), indicating it has a component of 1 in the x-direction and 3 in the y-direction. The second vector, \( \mathbf{v} \), points to the coordinates (2,0), indicating it has a component of 2 in the x-direction and 0 in the y-direction.
*Mathematical Explanation:*
The accompanying mathematical problem requires setting the vectors \( \mathbf{u} \) and \( \mathbf{v} \) such that the equation involving the dot product and magnitudes of the vectors holds true:
\[ \cos^{-1} \left( \frac{\mathbf{v} \cdot \mathbf{u}}{|\mathbf{v}| |\mathbf{u}|} \right) = \frac{\pi}{3} \]
In this case, \( \mathbf{v} \cdot \mathbf{u} \) represents the dot product of vectors \( \mathbf{v} \) and \( \mathbf{u} \), and \( |\mathbf{v}| \) and \( |\mathbf{u}| \) represent the magnitudes (lengths) of vectors \( \mathbf{v} \) and \( \mathbf{u} \) respectively.
The equation states that the angle between vectors \( \mathbf{v} \) and \( \mathbf{u} \) should be \( \frac{\pi}{3} \) radians, equivalent to 60 degrees.
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