To find the volume of the parallelepiped formed by the vectors **a**, **b**, and **c**, given that the volume formed by the vectors **a × b**, **b × c**, and **c × a** is 36 square units, we can follow these steps:
### Step 1: Understand the Volume of a Parallelepiped
The volume \( V \) of a parallelepiped formed by three vectors **a**, **b**, and **c** can be calculated using the scalar triple product:
\[
V = | \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) |
\]
### Step 2: Volume of the Given Vectors
We know that the volume formed by the vectors **a × b**, **b × c**, and **c × a** is given as 36 square units. This can be expressed as:
\[
V' = | (\mathbf{a} \times \mathbf{b}) \cdot ((\mathbf{b} \times \mathbf{c}) \times (\mathbf{c} \times \mathbf{a})) |
\]
However, we can simplify our calculations by using the properties of the scalar triple product.
### Step 3: Use the Properties of Cross and Dot Products
Using the identity for the scalar triple product, we can relate the two volumes:
\[
(\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{b} \times \mathbf{c}) \times (\mathbf{c} \times \mathbf{a}) = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \cdot \mathbf{c} \cdot \mathbf{a}
\]
This means that the volume formed by **a**, **b**, and **c** is related to the volume formed by **a × b**, **b × c**, and **c × a**.
### Step 4: Set Up the Equation
From the problem statement, we have:
\[
V' = 36
\]
Thus, we can conclude that:
\[
| \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) |^2 = 36
\]
### Step 5: Solve for the Volume of Vectors a, b, c
Taking the square root of both sides gives:
\[
| \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) | = \sqrt{36} = 6
\]
Thus, the volume formed by the vectors **a**, **b**, and **c** is:
\[
V = 6 \text{ cubic units}
\]
### Final Answer
The volume formed by the vectors **a**, **b**, and **c** is **6 cubic units**.
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