https://file.aiquickdraw.com/mathbot/file/1727448721084z1zhoy6h.jpg

To solve the problem of finding the angle of the bend at \( O \), we need to find the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \).

### Step-by-Step Solution:

1. **Determine the coordinates of points \( A \), \( O \), and \( B \):**
   - \( O \) is the origin \((0, 0, 0)\).
   - \( A \) is at the bottom-left corner of the room, so its coordinates are \((-2, 0, 0)\).
   - \( B \) is at the top-right corner, so its coordinates are \((0, 4, 2)\).

2. **Find vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \):**
   - \( \overrightarrow{OA} = A - O = (-2, 0, 0) \).
   - \( \overrightarrow{OB} = B - O = (0, 4, 2) \).

3. **Calculate the dot product \( \overrightarrow{OA} \cdot \overrightarrow{OB} \):**
   \[
   \overrightarrow{OA} \cdot \overrightarrow{OB} = (-2) \cdot 0 + 0 \cdot 4 + 0 \cdot 2 = 0
   \]

4. **Find the magnitudes of \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \):**
   - \( |\overrightarrow{OA}| = \sqrt{(-2)^2 + 0^2 + 0^2} = \sqrt{4} = 2 \).
   - \( |\overrightarrow{OB}| = \sqrt{0^2 + 4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \).

5. **Calculate the angle \( \theta \) between \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) using the dot product formula:**
   \[
   \cos \theta = \frac{\overrightarrow{OA} \cdot \overrightarrow{OB}}{|\overrightarrow{OA}| \times |\overrightarrow{OB}|}
   \]
   \[
   \cos \theta = \frac{0}{2 \times 2\sqrt{5}} = 0
   \]

6. **Find \( \theta \):**
   Since \( \cos \theta = 0 \), \( \theta = 90^\circ \).

Therefore, the angle of the bend at \( O \) is **90 degrees**.

Would you like any further details or have additional questions? Here are some related questions:

1. What is the significance of the dot product being zero in this context?
2. Can you find the distance between \( A \) and \( B \)?
3. How would the angle change if \( B \) were at a different height?
4. What is the shortest path a pipe can take between \( A \) and \( B \)?
5. How can we use the cross product to find the area of the triangle formed by \( A \), \( O \), and \( B \)?

**Tip:** Always verify the coordinates and the direction of the vectors carefully before calculating dot products or angles.

An engineer needs to bend a pipe at O so that it runs in a straight line from A to O, bends at O, and then runs in a straight line from O to B. What is the angle of the bend at O? The floor of the room is 2m × 4m, and the height is 2m.

Solve a 3D geometry problem to find the angle of a bend in a pipe between points A, O, and B in a rectangular room using vector analysis and dot product. Suitable for high school students learning about vectors and angles.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation