Volume of parallelopiped formed by vectors `vecaxxvecb, vecbxxvecc and veccxxveca` is `36` sq.units, then the volume of the parallelopiped formed by the vectors `veca,vecb` and `vecc` is.

Answer

To find the volume of the parallelepiped formed by the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), given that the volume of the parallelepiped formed by the vectors \(\vec{a} \times \vec{b}\), \(\vec{b} \times \vec{c}\), and \(\vec{c} \times \vec{a}\) is \(36\) square units, we can follow these steps:

### Step 1: Understand the Volume of Parallelepiped
The volume \(V\) of a parallelepiped formed by three vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) can be expressed as:
\[ V = |\vec{u} \cdot (\vec{v} \times \vec{w})| \]
For our case, we need to find:
\[ V = |\vec{a} \cdot (\vec{b} \times \vec{c})| \]

### Step 2: Relate Given Volume to Desired Volume
We are given that the volume of the parallelepiped formed by \(\vec{a} \times \vec{b}\), \(\vec{b} \times \vec{c}\), and \(\vec{c} \times \vec{a}\) is \(36\). We can express this volume as:
\[ V' = |(\vec{a} \times \vec{b}) \cdot ((\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}))| \]

### Step 3: Use Vector Triple Product Identity
Using the vector triple product identity:
\[ \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z} \]
we can simplify the expression for \(V'\).

### Step 4: Substitute and Simplify
Let \(\vec{R} = \vec{b} \times \vec{c}\). Then:
\[ V' = |(\vec{a} \times \vec{b}) \cdot (\vec{R} \times (\vec{c} \times \vec{a}))| \]
Using the vector triple product:
\[ \vec{R} \times (\vec{c} \times \vec{a}) = (\vec{R} \cdot \vec{a}) \vec{c} - (\vec{R} \cdot \vec{c}) \vec{a} \]
Substituting this back into the volume expression.

### Step 5: Calculate the Volume
From the properties of determinants and the fact that the volume of a parallelepiped is invariant under cyclic permutations, we can conclude:
\[ V = \sqrt{V'^2} = \sqrt{36} = 6 \]

### Final Answer
Thus, the volume of the parallelepiped formed by the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) is:
\[ \boxed{6} \]