Given that a curve is moving directly upward (positive z direction) when t = 4, and that the normal vector at t= 4 is in the direction parallel to < 4, − 10,0 > . (Round answers to two decimal places.) (a) Find the i component of N(4), the unit normal vector. (b) Find the j component of N(4), the unit normal vector. (c) Find the k component of N(4), the unit normal vector. (d) Find the i component of B(4), the binormal vector. (e) Find the j component of B(4), the binormal vector. (f) Find the k component of B(4), the binormal vector.
Given that a curve is moving directly upward (positive z direction) when t = 4, and that the normal vector at t= 4 is in the direction parallel to < 4, − 10,0 > . (Round answers to two decimal places.) (a) Find the i component of N(4), the unit normal vector. (b) Find the j component of N(4), the unit normal vector. (c) Find the k component of N(4), the unit normal vector. (d) Find the i component of B(4), the binormal vector. (e) Find the j component of B(4), the binormal vector. (f) Find the k component of B(4), the binormal vector.
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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Question
![### Problem Statement
Given that a curve is moving directly upward (positive \( z \)-direction) when \( t = 4 \), and that the normal vector at \( t = 4 \) is in the direction parallel to \( \langle 4, -10, 0 \rangle \). (Round answers to two decimal places.)
1. **Find the \( i \) component of \( \mathbf{N}(4) \), the unit normal vector.**
\[
\boxed{}
\]
2. **Find the \( j \) component of \( \mathbf{N}(4) \), the unit normal vector.**
\[
\boxed{}
\]
3. **Find the \( k \) component of \( \mathbf{N}(4) \), the unit normal vector.**
\[
\boxed{}
\]
4. **Find the \( i \) component of \( \mathbf{B}(4) \), the binormal vector.**
\[
\boxed{}
\]
5. **Find the \( j \) component of \( \mathbf{B}(4) \), the binormal vector.**
\[
\boxed{}
\]
6. **Find the \( k \) component of \( \mathbf{B}(4) \), the binormal vector.**
\[
\boxed{}
\]
### Explanation
- **Unit Normal Vector (\( \mathbf{N}(t) \))**: A vector that is perpendicular to the tangent vector of the curve at a given point and has a magnitude of 1.
- **Binormal Vector (\( \mathbf{B}(t) \))**: A vector that is perpendicular to both the tangent vector and the normal vector at a point on the curve, often achieved by taking the cross product of the tangent and normal vectors.
The given vector \( \langle 4, -10, 0 \rangle \) needs to be normalized to achieve the unit normal vector.
### Steps to Solution
1. **Normalize the Normal Vector**:
\[
\mathbf{N}(t) = \frac{\langle 4, -10, 0 \rangle}{\|\langle 4, -10, 0 \rangle\|} = \frac{\langle 4, -10, 0 \r](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe4962ae7-1cb2-4276-8f6a-b9f851b67289%2Fd5749627-1ea8-468e-b72a-4d6825643b10%2Fhfvzbpe_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Given that a curve is moving directly upward (positive \( z \)-direction) when \( t = 4 \), and that the normal vector at \( t = 4 \) is in the direction parallel to \( \langle 4, -10, 0 \rangle \). (Round answers to two decimal places.)
1. **Find the \( i \) component of \( \mathbf{N}(4) \), the unit normal vector.**
\[
\boxed{}
\]
2. **Find the \( j \) component of \( \mathbf{N}(4) \), the unit normal vector.**
\[
\boxed{}
\]
3. **Find the \( k \) component of \( \mathbf{N}(4) \), the unit normal vector.**
\[
\boxed{}
\]
4. **Find the \( i \) component of \( \mathbf{B}(4) \), the binormal vector.**
\[
\boxed{}
\]
5. **Find the \( j \) component of \( \mathbf{B}(4) \), the binormal vector.**
\[
\boxed{}
\]
6. **Find the \( k \) component of \( \mathbf{B}(4) \), the binormal vector.**
\[
\boxed{}
\]
### Explanation
- **Unit Normal Vector (\( \mathbf{N}(t) \))**: A vector that is perpendicular to the tangent vector of the curve at a given point and has a magnitude of 1.
- **Binormal Vector (\( \mathbf{B}(t) \))**: A vector that is perpendicular to both the tangent vector and the normal vector at a point on the curve, often achieved by taking the cross product of the tangent and normal vectors.
The given vector \( \langle 4, -10, 0 \rangle \) needs to be normalized to achieve the unit normal vector.
### Steps to Solution
1. **Normalize the Normal Vector**:
\[
\mathbf{N}(t) = \frac{\langle 4, -10, 0 \rangle}{\|\langle 4, -10, 0 \rangle\|} = \frac{\langle 4, -10, 0 \r
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