If the given three vectors are coplanar, then find the value of x. A = i-2j + 3k, B = xj + 3k, C = 7i+3j - 11k
If the given three vectors are coplanar, then find the value of x. A = i-2j + 3k, B = xj + 3k, C = 7i+3j - 11k
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Problem 4:
If the given three vectors are coplanar, find the value of \( x \).
\[ A = \mathbf{i} - 2\mathbf{j} + 3\mathbf{k} \]
\[ B = x\mathbf{j} + 3\mathbf{k} \]
\[ C = 7\mathbf{i} + 3\mathbf{j} - 11\mathbf{k} \]
### Options:
a. \( F = x^2 \sin(5y) \)
b. \( F = x^4 yz \)
c. \( F = x^3 e^{xy} + e^{2x} \)
---
To determine the value of \( x \) for which the vectors \( A \), \( B \), and \( C \) are coplanar, determine if these vectors lie in the same plane. This can be using the scalar triple product, which states that vectors \( A \), \( B \), and \( C \) are coplanar if their scalar triple product is zero:
\[ \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = 0 \]
Calculate the cross product \( \mathbf{B} \times \mathbf{C} \), then the dot product \( \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) \), and solve for \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a780900-9b69-4fe9-9f43-185066935266%2F98456a52-f124-4baf-949e-1c2a1140f1cf%2F3babo97_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 4:
If the given three vectors are coplanar, find the value of \( x \).
\[ A = \mathbf{i} - 2\mathbf{j} + 3\mathbf{k} \]
\[ B = x\mathbf{j} + 3\mathbf{k} \]
\[ C = 7\mathbf{i} + 3\mathbf{j} - 11\mathbf{k} \]
### Options:
a. \( F = x^2 \sin(5y) \)
b. \( F = x^4 yz \)
c. \( F = x^3 e^{xy} + e^{2x} \)
---
To determine the value of \( x \) for which the vectors \( A \), \( B \), and \( C \) are coplanar, determine if these vectors lie in the same plane. This can be using the scalar triple product, which states that vectors \( A \), \( B \), and \( C \) are coplanar if their scalar triple product is zero:
\[ \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = 0 \]
Calculate the cross product \( \mathbf{B} \times \mathbf{C} \), then the dot product \( \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) \), and solve for \( x \).
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