Suppose ABC is a right angled triangle with `angleC=pi//2`. If `abs(AB)=5," then "AB*AC+BC*BA+CA*CB=`________

Answer

To solve the problem, we need to calculate the expression \( AB \cdot AC + BC \cdot BA + CA \cdot CB \) given that \( \angle C = \frac{\pi}{2} \) and \( |AB| = 5 \).

### Step-by-Step Solution:

1. **Understanding the Triangle**:
Since \( ABC \) is a right-angled triangle with \( \angle C = \frac{\pi}{2} \), we know that \( AB \) is the hypotenuse, and \( AC \) and \( BC \) are the two legs of the triangle.

2. **Using the Dot Product**:
The dot product of two vectors \( \vec{x} \) and \( \vec{y} \) is given by:
\[ \vec{x} \cdot \vec{y} = |\vec{x}| |\vec{y}| \cos(\theta) \]
where \( \theta \) is the angle between the two vectors.

3. **Calculating Each Dot Product**:
- For \( AB \cdot AC \):
\[ AB \cdot AC = |AB| |AC| \cos(\alpha) \]
- For \( BC \cdot BA \):
\[ BC \cdot BA = |BC| |AB| \cos(\beta) \]
- For \( CA \cdot CB \):
Since \( CA \) and \( CB \) are perpendicular (as \( \angle C = \frac{\pi}{2} \)):
\[ CA \cdot CB = 0 \]

4. **Substituting Values**:
Therefore, the expression simplifies to:
\[ AB \cdot AC + BC \cdot BA + CA \cdot CB = AB \cdot AC + BC \cdot BA \]

5. **Using the Right Triangle Properties**:
Given \( |AB| = 5 \), we can express \( |AC| \) and \( |BC| \) in terms of \( \alpha \) and \( \beta \) (the angles opposite to sides \( AC \) and \( BC \)):
- From the definition of cosine in a right triangle:
\[ |AC| = |AB| \cos(\alpha) = 5 \cos(\alpha) \]
\[ |BC| = |AB| \sin(\alpha) = 5 \sin(\alpha) \]

6. **Substituting Back**:
Now substituting back into our expression:
\[ AB \cdot AC = 5 \cdot (5 \cos(\alpha)) \cos(\alpha) = 25 \cos^2(\alpha) \]
\[ BC \cdot BA = (5 \sin(\alpha)) \cdot 5 \cos(\beta) = 25 \sin(\alpha) \cos(\beta) \]
But since \( \beta = \frac{\pi}{2} - \alpha \), we have \( \cos(\beta) = \sin(\alpha) \):
\[ BC \cdot BA = 25 \sin^2(\alpha) \]

7. **Final Calculation**:
Therefore, the expression becomes:
\[ 25 \cos^2(\alpha) + 25 \sin^2(\alpha) = 25 (\cos^2(\alpha) + \sin^2(\alpha)) = 25 \cdot 1 = 25 \]

### Final Answer:
Thus, the value of \( AB \cdot AC + BC \cdot BA + CA \cdot CB \) is \( \boxed{25} \).