Find a unit vector orthogonal to both u and v. u = −5i + 3j - 2k 11-31+ 1 j+_¹_k 10 V =
Find a unit vector orthogonal to both u and v. u = −5i + 3j - 2k 11-31+ 1 j+_¹_k 10 V =
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:**
Find a unit vector orthogonal to both **u** and **v**.
**Vectors Given:**
\[
\mathbf{u} = -5\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}
\]
\[
\mathbf{v} = \frac{1}{2}\mathbf{i} - \frac{3}{4}\mathbf{j} + \frac{1}{10}\mathbf{k}
\]
**Explanation:**
To find a unit vector orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\), you need to calculate the cross product \(\mathbf{u} \times \mathbf{v}\) and then normalize it to obtain a unit vector.
The cross product \(\mathbf{u} \times \mathbf{v}\) gives a vector that is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\).
1. **Cross Product Calculation:**
Use the determinant formula for cross products:
\[
\mathbf{u} \times \mathbf{v} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
-5 & 3 & -2 \\
\frac{1}{2} & -\frac{3}{4} & \frac{1}{10}
\end{vmatrix}
\]
2. **Normalization:**
Calculate the magnitude of the resulting vector from the cross product and divide each component of the vector by this magnitude to obtain the unit vector.
This unit vector will represent the direction that is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4b64424-2f1e-400b-a2f3-ccc24d7d74e8%2Fd07ccb5e-1be7-4d3b-a4b8-82b4a4fa330f%2Fzyffz7_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find a unit vector orthogonal to both **u** and **v**.
**Vectors Given:**
\[
\mathbf{u} = -5\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}
\]
\[
\mathbf{v} = \frac{1}{2}\mathbf{i} - \frac{3}{4}\mathbf{j} + \frac{1}{10}\mathbf{k}
\]
**Explanation:**
To find a unit vector orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\), you need to calculate the cross product \(\mathbf{u} \times \mathbf{v}\) and then normalize it to obtain a unit vector.
The cross product \(\mathbf{u} \times \mathbf{v}\) gives a vector that is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\).
1. **Cross Product Calculation:**
Use the determinant formula for cross products:
\[
\mathbf{u} \times \mathbf{v} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
-5 & 3 & -2 \\
\frac{1}{2} & -\frac{3}{4} & \frac{1}{10}
\end{vmatrix}
\]
2. **Normalization:**
Calculate the magnitude of the resulting vector from the cross product and divide each component of the vector by this magnitude to obtain the unit vector.
This unit vector will represent the direction that is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\).
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