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...
Related questions
Question
#7
Please solve on paper.
![**Problem 7:** For what values of \( c \) are the vectors \([-8, c, 4]\) and \([c, c^2, c]\) orthogonal?
**Explanation:**
To determine the values of \( c \) for which the vectors are orthogonal, we need to find when their dot product is zero. The dot product of two vectors \([a_1, a_2, a_3]\) and \([b_1, b_2, b_3]\) is calculated as:
\[ a_1 \cdot b_1 + a_2 \cdot b_2 + a_3 \cdot b_3 = 0 \]
Applying this to the given vectors:
\[ (-8) \cdot c + c \cdot c^2 + 4 \cdot c = 0 \]
Simplifying:
\[ -8c + c^3 + 4c = 0 \]
\[ c^3 - 4c = 0 \]
Factor out \( c \) from the equation:
\[ c(c^2 - 4) = 0 \]
This gives:
\[ c = 0 \quad \text{or} \quad c^2 - 4 = 0 \]
Solve \( c^2 - 4 = 0 \):
\[ c^2 = 4 \]
\[ c = 2 \quad \text{or} \quad c = -2 \]
Thus, the vectors are orthogonal for \( c = 0, 2, \) or \(-2\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F19245b5f-ae26-4cd9-93a9-b35608880650%2F455481bf-c4b0-42ab-9b6d-56c4448ba4b3%2Fzx3ggf3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 7:** For what values of \( c \) are the vectors \([-8, c, 4]\) and \([c, c^2, c]\) orthogonal?
**Explanation:**
To determine the values of \( c \) for which the vectors are orthogonal, we need to find when their dot product is zero. The dot product of two vectors \([a_1, a_2, a_3]\) and \([b_1, b_2, b_3]\) is calculated as:
\[ a_1 \cdot b_1 + a_2 \cdot b_2 + a_3 \cdot b_3 = 0 \]
Applying this to the given vectors:
\[ (-8) \cdot c + c \cdot c^2 + 4 \cdot c = 0 \]
Simplifying:
\[ -8c + c^3 + 4c = 0 \]
\[ c^3 - 4c = 0 \]
Factor out \( c \) from the equation:
\[ c(c^2 - 4) = 0 \]
This gives:
\[ c = 0 \quad \text{or} \quad c^2 - 4 = 0 \]
Solve \( c^2 - 4 = 0 \):
\[ c^2 = 4 \]
\[ c = 2 \quad \text{or} \quad c = -2 \]
Thus, the vectors are orthogonal for \( c = 0, 2, \) or \(-2\).
Expert Solution
Step 1
For orthogonal,dot product of vectors is equal to zero.
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning