Find the volume of a parallelopiped having three coterminus vectors of equal magnitude `|a|` and equal inclination `theta` with each other.

Explanation

To find the volume of a parallelepiped formed by three coterminus vectors of equal magnitude \( |a| \) and equal inclination \( \theta \) with each other, we can follow these steps:

### Step 1: Define the vectors
Let the three vectors be \( \vec{A}, \vec{B}, \vec{C} \) such that:
- \( |\vec{A}| = |\vec{B}| = |\vec{C}| = |a| \)
- The angle between any two vectors is \( \theta \).

### Step 2: Use the formula for volume
The volume \( V \) of the parallelepiped formed by the vectors \( \vec{A}, \vec{B}, \vec{C} \) can be calculated using the scalar triple product:
\[ V = |\vec{A} \cdot (\vec{B} \times \vec{C})| \]

### Step 3: Calculate \( \vec{B} \times \vec{C} \)
To find \( \vec{B} \times \vec{C} \), we can use the formula:
\[ |\vec{B} \times \vec{C}| = |\vec{B}| |\vec{C}| \sin \theta \]
Since \( |\vec{B}| = |\vec{C}| = |a| \), we have:
\[ |\vec{B} \times \vec{C}| = |a| |a| \sin \theta = |a|^2 \sin \theta \]

### Step 4: Calculate \( \vec{A} \cdot (\vec{B} \times \vec{C}) \)
Now, we need to compute \( \vec{A} \cdot (\vec{B} \times \vec{C}) \):
\[ \vec{A} \cdot (\vec{B} \times \vec{C}) = |\vec{A}| |\vec{B} \times \vec{C}| \cos \phi \]
where \( \phi \) is the angle between \( \vec{A} \) and \( \vec{B} \times \vec{C} \).

### Step 5: Find \( \cos \phi \)
The angle \( \phi \) can be determined using the properties of the vectors. Since all vectors have equal inclination \( \theta \), we can express \( \cos \phi \) in terms of \( \theta \).

Using the relationship of the angles, we can find that:
\[ \cos \phi = \sqrt{1 - \left(\frac{1}{2} \cdot \cos \theta\right)^2} \]

### Step 6: Substitute and simplify
Now substituting back, we have:
\[ V = |\vec{A}| \cdot |a|^2 \sin \theta \cdot \cos \phi \]
Substituting \( |\vec{A}| = |a| \):
\[ V = |a|^3 \sin \theta \cdot \cos \phi \]

### Step 7: Final expression for volume
After substituting and simplifying, we find:
\[ V = |a|^3 \sin \theta \cdot \sqrt{1 - \left(\frac{1}{2} \cdot \cos \theta\right)^2} \]

### Conclusion
Thus, the volume of the parallelepiped formed by the three coterminus vectors of equal magnitude \( |a| \) and equal inclination \( \theta \) is given by:
\[ V = |a|^3 \sin \theta \cdot \sqrt{1 - \left(\frac{1}{2} \cdot \cos \theta\right)^2} \]