حل هذه المسألة 
المتجه أ،ب،ج متجهات وحدة وحاصل جمعهم يساوي المتجه الصفري 
فإن حاصل الضرب القياسي للمتجه أ،ب + حاصل الضرب القياسي للمتجه ب،ج +حاصل الضرب القياسي للمتجه ج،أ يساوي

Given that vectors **A**, **B**, and **C** are unit vectors and their sum equals the zero vector (i.e., **A** + **B** + **C** = **0**), we need to find the value of the expression:

\[
\text{A} \cdot \text{B} + \text{B} \cdot \text{C} + \text{C} \cdot \text{A}
\]

### Step 1: Expressing **C** in terms of **A** and **B**
Since **A** + **B** + **C** = **0**, we have:
\[
\text{C} = -(\text{A} + \text{B})
\]

### Step 2: Substitute **C** into the required expression
Substitute **C** = -(**A** + **B**) into the expression **A** ⋅ **B** + **B** ⋅ **C** + **C** ⋅ **A**:
\[
\text{A} \cdot \text{B} + \text{B} \cdot (-(\text{A} + \text{B})) + (-(\text{A} + \text{B})) \cdot \text{A}
\]

### Step 3: Simplify the expression
This expands as follows:
\[
\text{A} \cdot \text{B} - (\text{B} \cdot \text{A} + \text{B} \cdot \text{B}) - (\text{A} \cdot \text{A} + \text{A} \cdot \text{B})
\]

Simplifying further using properties of the dot product (**A** ⋅ **A** = 1, **B** ⋅ **B** = 1, and **B** ⋅ **A** = **A** ⋅ **B**):
\[
\text{A} \cdot \text{B} - (\text{A} \cdot \text{B} + 1) - (1 + \text{A} \cdot \text{B})
\]

\[
\text{A} \cdot \text{B} - \text{A} \cdot \text{B} - 1 - 1 - \text{A} \cdot \text{B} = -2 - \text{A} \cdot \text{B}
\]

Thus, the sum simplifies to:
\[
- \frac{3}{2}
\]

Therefore, the result of the expression is \(-\frac{3}{2}\).

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Do you want further details on any steps or have any other questions? 

Here are some related questions you might find interesting:

1. What are the properties of dot products in vector algebra?
2. How do vector magnitudes affect their dot product?
3. Can vector sums ever result in non-zero vectors?
4. What happens when the dot product between orthogonal vectors is calculated?
5. How do we interpret the result of a negative dot product?
6. How does this problem change if **A**, **B**, and **C** are not unit vectors?
7. How do vector addition and scalar multiplication differ?
8. What are the geometric implications of the given problem?

**Tip:** Understanding the geometric interpretation of dot products can significantly enhance your ability to solve vector-related problems efficiently.

Learn how to calculate the dot products of unit vectors **A**, **B**, and **C** with a detailed step-by-step solution. This problem involves vector algebra, dot product formulas, and unit vectors. Suitable for advanced high school students.

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