Suppose a, b, c are three non-coplanar vectors. Suppose
`Delta=|{:(a*a, a* b, a*c), (b*a, b*b, b*c), (c*a, c*b, c*c):}|`
If `Delta=[a b c]^(r)" then " r=` _______

Answer

To solve the problem, we need to evaluate the determinant \(\Delta\) defined as:

\[ \Delta = \begin{vmatrix} a \cdot a & a \cdot b & a \cdot c \\ b \cdot a & b \cdot b & b \cdot c \\ c \cdot a & c \cdot b & c \cdot c \end{vmatrix} \]

We are given that \(\Delta = [a \, b \, c]^r\) and we need to find the value of \(r\).

### Step 1: Understanding the Determinant
The determinant \(\Delta\) consists of dot products of the vectors \(a\), \(b\), and \(c\). The elements of the matrix are:

- \(a \cdot a = |a|^2\)
- \(a \cdot b\)
- \(a \cdot c\)
- \(b \cdot a\)
- \(b \cdot b = |b|^2\)
- \(b \cdot c\)
- \(c \cdot a\)
- \(c \cdot b\)
- \(c \cdot c = |c|^2\)

### Step 2: Recognizing the Structure
This determinant can be recognized as a Gram determinant, which is related to the volumes of the parallelepiped formed by the vectors \(a\), \(b\), and \(c\). The volume \(V\) of the parallelepiped is given by:

\[ V = |[a \, b \, c]| = \sqrt{\Delta} \]

### Step 3: Evaluating the Determinant
The determinant \(\Delta\) can be expressed as:

\[ \Delta = |a|^2 |b|^2 |c|^2 - (a \cdot b)^2 |c|^2 - (a \cdot c)^2 |b|^2 - (b \cdot c)^2 |a|^2 + 2(a \cdot b)(a \cdot c)(b \cdot c) \]

### Step 4: Relating \(\Delta\) to \([a \, b \, c]^r\)
From the properties of determinants, we know that:

\[ \Delta = |[a \, b \, c]|^2 \]

Thus, we can equate:

\[ \Delta = |[a \, b \, c]|^2 = [a \, b \, c]^2 \]

### Step 5: Finding \(r\)
Since we have established that \(\Delta = [a \, b \, c]^2\), we can conclude that:

\[ \Delta = [a \, b \, c]^r \implies [a \, b \, c]^2 = [a \, b \, c]^r \]

This implies that:

\[ r = 2 \]

### Final Answer
Thus, the value of \(r\) is:

\[ \boxed{2} \]