In a regular tetrahedron, let `theta` be angle between any edge and a face not containing the edge. Then the value of `cos^(2)theta` is

Options

A. `1//6`
B. `1//9`
C. `1//3`
D. none of these

Answer

To solve the problem, we need to find the value of \( \cos^2 \theta \) where \( \theta \) is the angle between any edge of a regular tetrahedron and a face not containing that edge.

### Step-by-Step Solution:

1. **Understanding the Geometry of a Regular Tetrahedron**:
A regular tetrahedron has four vertices, and each vertex is connected to the other three vertices by edges. The faces of the tetrahedron are equilateral triangles.

2. **Labeling the Vertices**:
Let the vertices of the tetrahedron be labeled as \( O, A, B, C \). The edges are \( OA, OB, OC, AB, AC, BC \).

3. **Identifying the Edge and the Face**:
We will consider the edge \( OA \) and the face \( OBC \). We need to find the angle \( \theta \) between the edge \( OA \) and the plane formed by the face \( OBC \).

4. **Finding the Normal Vector of the Face**:
To find the angle \( \theta \), we first need to determine the normal vector to the face \( OBC \). The vectors \( \overrightarrow{OB} \) and \( \overrightarrow{OC} \) can be represented as unit vectors in the direction of \( B \) and \( C \).

5. **Using the Cross Product**:
The normal vector \( \mathbf{n} \) to the face \( OBC \) can be found using the cross product:
\[ \mathbf{n} = \overrightarrow{OB} \times \overrightarrow{OC} \]

6. **Calculating the Angle**:
The angle \( \theta \) between the edge \( OA \) and the normal vector \( \mathbf{n} \) can be found using the dot product:
\[ \cos \theta = \frac{\overrightarrow{OA} \cdot \mathbf{n}}{|\overrightarrow{OA}| |\mathbf{n}|} \]

7. **Finding the Magnitudes**:
Since \( OA, OB, OC \) are edges of a regular tetrahedron, they are equal in length. Let’s denote the length of each edge as \( a \). Thus, \( |\overrightarrow{OA}| = a \) and \( |\mathbf{n}| = \sqrt{3} \cdot \frac{a^2}{2} \) (since the area of the triangle is \( \frac{\sqrt{3}}{4} a^2 \)).

8. **Substituting Values**:
From the geometry of the tetrahedron, we know that the angle between any two edges is \( 60^\circ \). Therefore, we can substitute the values into the equation:
\[ \cos \theta = \frac{1}{\sqrt{3}} \]

9. **Finding \( \cos^2 \theta \)**:
Now, we can find \( \cos^2 \theta \):
\[ \cos^2 \theta = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \]

### Final Answer:
Thus, the value of \( \cos^2 \theta \) is \( \frac{1}{3} \).