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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q4
11.4
![**Finding a Unit Vector Orthogonal to Two Given Vectors**
To solve the problem, we need to identify a unit vector that is orthogonal to both vectors **u** and **v**.
Given:
\[ \mathbf{u} = \langle -8, -6, 4 \rangle \]
\[ \mathbf{v} = \langle 17, -18, -1 \rangle \]
### Steps to Find the Orthogonal Vector:
1. **Compute the Cross Product** \( \mathbf{u} \times \mathbf{v} \):
The cross product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) will give a vector that is orthogonal to both. This is calculated using the determinant of a matrix formed by unit vectors and the two given vectors.
2. **Normalize the Resulting Vector**:
Calculate the magnitude of the vector obtained from the cross product and then divide each component by this magnitude to get the unit vector.
### Details:
- **Cross Product Formula**:
For vectors \(\mathbf{u} = \langle u_1, u_2, u_3 \rangle\) and \(\mathbf{v} = \langle v_1, v_2, v_3 \rangle\), the cross product \(\mathbf{u} \times \mathbf{v}\) is given by:
\[
\mathbf{u} \times \mathbf{v} = \left| \begin{matrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
u_1 & u_2 & u_3 \\
v_1 & v_2 & v_3
\end{matrix} \right|
\]
This expands to:
\[
(u_2 v_3 - u_3 v_2) \mathbf{i} - (u_1 v_3 - u_3 v_1) \mathbf{j} + (u_1 v_2 - u_2 v_1) \mathbf{k}
\]
- **Example Computation**:
Using the vectors given:
\(
\mathbf{u} = \langle -8, -6, 4 \rangle \)
and
\(
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd09d3726-e6d4-40fb-b56b-e32001dd8288%2F85d63ce1-877a-4e42-8f88-daea64d2d917%2Fmgoye4m_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding a Unit Vector Orthogonal to Two Given Vectors**
To solve the problem, we need to identify a unit vector that is orthogonal to both vectors **u** and **v**.
Given:
\[ \mathbf{u} = \langle -8, -6, 4 \rangle \]
\[ \mathbf{v} = \langle 17, -18, -1 \rangle \]
### Steps to Find the Orthogonal Vector:
1. **Compute the Cross Product** \( \mathbf{u} \times \mathbf{v} \):
The cross product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) will give a vector that is orthogonal to both. This is calculated using the determinant of a matrix formed by unit vectors and the two given vectors.
2. **Normalize the Resulting Vector**:
Calculate the magnitude of the vector obtained from the cross product and then divide each component by this magnitude to get the unit vector.
### Details:
- **Cross Product Formula**:
For vectors \(\mathbf{u} = \langle u_1, u_2, u_3 \rangle\) and \(\mathbf{v} = \langle v_1, v_2, v_3 \rangle\), the cross product \(\mathbf{u} \times \mathbf{v}\) is given by:
\[
\mathbf{u} \times \mathbf{v} = \left| \begin{matrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
u_1 & u_2 & u_3 \\
v_1 & v_2 & v_3
\end{matrix} \right|
\]
This expands to:
\[
(u_2 v_3 - u_3 v_2) \mathbf{i} - (u_1 v_3 - u_3 v_1) \mathbf{j} + (u_1 v_2 - u_2 v_1) \mathbf{k}
\]
- **Example Computation**:
Using the vectors given:
\(
\mathbf{u} = \langle -8, -6, 4 \rangle \)
and
\(
\
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