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: Understand the Volume Formula
The volume \(V\) of a parallelepiped formed by three vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) can be expressed using the scalar triple product:
\[ V = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| \]

### Step 2: Set Up the Vectors
Since the vectors are coterminus and have equal magnitudes, we can denote:
\[ |\mathbf{a}| = |\mathbf{b}| = |\mathbf{c}| = |a| \]
Let \(\mathbf{a} = \mathbf{b} = \mathbf{c} = \mathbf{a}\).

### Step 3: Calculate the Scalar Triple Product
Using the property of the scalar triple product, we have:
\[ V^2 = (\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}))^2 \]
Substituting \(\mathbf{b} = \mathbf{a}\) and \(\mathbf{c} = \mathbf{a}\):
\[ V^2 = (\mathbf{a} \cdot (\mathbf{a} \times \mathbf{a}))^2 \]
Since the cross product of any vector with itself is zero:
\[ \mathbf{a} \times \mathbf{a} = \mathbf{0} \]
Thus,
\[ V^2 = (\mathbf{a} \cdot \mathbf{0})^2 = 0 \]
This indicates that the volume is zero when all three vectors are the same.

### Step 4: Consider the Angles Between Vectors
To find the volume when the vectors are not identical but have equal angles \(\theta\) between them, we can use the formula for the volume of a parallelepiped in terms of the angles between the vectors:
\[ V = |a|^3 \sqrt{1 - 3\cos^2\theta + 2\cos^3\theta} \]
This formula arises from the determinant of the matrix formed by the vectors and their angles.

### Step 5: Final Expression for Volume
Thus, the volume of the parallelepiped can be expressed as:
\[ V = |a|^3 \sqrt{(1 - \cos\theta)^2(1 + 2\cos\theta)} \]

### Conclusion
The volume of the parallelepiped formed by three coterminus vectors of equal magnitude \(|a|\) and equal inclination \(\theta\) with each other is:
\[ V = |a|^3 \sqrt{(1 - \cos\theta)^2(1 + 2\cos\theta)} \]