https://file.aiquickdraw.com/mathbot/file/1726516227879gvsr5ole.jpg

The question asks to find the angles of the triangle formed by the points \((2, 1, 2)\), \((4, -1, 5)\), and \((7, 1, -2)\) in degrees.

We can approach this problem by first calculating the vectors corresponding to the sides of the triangle, then using the dot product to find the angles between these vectors.

Let the vertices be labeled as follows:
- \( A(2, 1, 2) \),
- \( B(4, -1, 5) \),
- \( C(7, 1, -2) \).

### Steps:

1. **Calculate the vectors representing the sides:**
   - Vector \( \overrightarrow{AB} = B - A = (4 - 2, -1 - 1, 5 - 2) = (2, -2, 3) \),
   - Vector \( \overrightarrow{AC} = C - A = (7 - 2, 1 - 1, -2 - 2) = (5, 0, -4) \),
   - Vector \( \overrightarrow{BC} = C - B = (7 - 4, 1 + 1, -2 - 5) = (3, 2, -7) \).

2. **Use the dot product formula to find the cosine of the angles:**
   The formula for the dot product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is:
   \[
   \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos \theta
   \]
   where \( \theta \) is the angle between the vectors, and \( |\mathbf{u}| \) and \( |\mathbf{v}| \) are the magnitudes of the vectors.

   We can rearrange the formula to solve for \( \theta \):
   \[
   \theta = \cos^{-1} \left( \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|} \right)
   \]

3. **Calculate the angles at each vertex.**

I will now calculate the vectors' magnitudes, dot products, and angles for each of the vertices.The angles of the triangle with vertices \((2, 1, 2)\), \((4, -1, 5)\), and \((7, 1, -2)\) are approximately:

- Angle at vertex \( A \): \( 94.34^\circ \),
- Angle at vertex \( B \): \( 125.82^\circ \),
- Angle at vertex \( C \): \( 31.48^\circ \).

Would you like more details or further explanations?

Here are 5 related questions to explore further:
1. How can you use the Law of Cosines to verify these angles?
2. How would the process differ for

Find, in degrees, the angles of the triangle with vertices (2, 1, 2), (4, -1, 5), and (7, 1, -2).

Learn how to calculate the angles of a triangle in 3D space with vertices at (2, 1, 2), (4, -1, 5), and (7, 1, -2). This problem involves vector subtraction, dot product calculations, and solving for angles using inverse cosine. Suitable for high school and undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation