**Finding a Unit Vector Orthogonal to Both Vectors****Problem Statement:**Find a unit vector orthogonal to both **u** and **v**.**Given Vectors:**\[ \mathbf{u} = 3\mathbf{i} + 5\mathbf{j} + 2\mathbf{k} \]\[ \mathbf{v} = \frac{1}{2}\mathbf{i} - \frac{3}{4}\mathbf{j} + \frac{1}{10}\mathbf{k} \]**Solution:**1. **Cross Product:** Calculate the cross product \(\mathbf{u} \times \mathbf{v}\) to find a vector orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\). \[ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 3 & 5 & 2 \\ \frac{1}{2} & -\frac{3}{4} & \frac{1}{10} \end{vmatrix} \] \[ \mathbf{u} \times \mathbf{v} = \mathbf{i} \left( 5 \cdot \frac{1}{10} - 2 \cdot \left( -\frac{3}{4} \right) \right) - \mathbf{j} \left( 3 \cdot \frac{1}{10} - 2 \cdot \left( \frac{1}{2} \right) \right) + \mathbf{k} \left( 3 \cdot \left( -\frac{3}{4} \right) - 5 \cdot \left( \frac{1}{2} \right) \right) \] \[ \mathbf{u} \times \mathbf{v} = \mathbf{i} \left( \frac{5}{10} + \frac{6}{4} \right) - \mathbf{j} \left( \frac{3}{10} - 1 \right) + \mathbf{k} \left( -\frac{9}{4} - \frac{5

Given the vectors u→=3i→+5j→+2k→, v→=12i→-34j→+110k→. To find: A unit vector orthogonal to both u→…

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Find a unit vector orthogonal to both u and v. u = 3i + 5j + 2k V= 위해 + -k 10

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

**Finding a Unit Vector Orthogonal to Both Vectors**

**Problem Statement:**
Find a unit vector orthogonal to both **u** and **v**.

**Given Vectors:**
\[ \mathbf{u} = 3\mathbf{i} + 5\mathbf{j} + 2\mathbf{k} \]
\[ \mathbf{v} = \frac{1}{2}\mathbf{i} - \frac{3}{4}\mathbf{j} + \frac{1}{10}\mathbf{k} \]

**Solution:**

1. **Cross Product:**
Calculate the cross product \(\mathbf{u} \times \mathbf{v}\) to find a vector orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\).

\[
\mathbf{u} \times \mathbf{v} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
3 & 5 & 2 \\
\frac{1}{2} & -\frac{3}{4} & \frac{1}{10}
\end{vmatrix}
\]

\[
\mathbf{u} \times \mathbf{v} = \mathbf{i} \left( 5 \cdot \frac{1}{10} - 2 \cdot \left( -\frac{3}{4} \right) \right)
- \mathbf{j} \left( 3 \cdot \frac{1}{10} - 2 \cdot \left( \frac{1}{2} \right) \right)
+ \mathbf{k} \left( 3 \cdot \left( -\frac{3}{4} \right) - 5 \cdot \left( \frac{1}{2} \right) \right)
\]

\[
\mathbf{u} \times \mathbf{v} = \mathbf{i} \left( \frac{5}{10} + \frac{6}{4} \right)
- \mathbf{j} \left( \frac{3}{10} - 1 \right)
+ \mathbf{k} \left( -\frac{9}{4} - \frac{5

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