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give a detailed answer to this math problem

The image contains a problem written in Chinese. The math problem states the following:

**证明：向量 a 与 (a·c)b - (a·b)c 垂直**

This translates to:

**Prove that vector \( a \) is perpendicular to the vector \( (a \cdot c)b - (a \cdot b)c \).**

### Solution:

To prove that two vectors are perpendicular, we need to show that their dot product is zero. In this case, we need to show that:

\[
a \cdot \left[ (a \cdot c)b - (a \cdot b)c \right] = 0
\]

Let's break this down step by step.

### Step 1: Distribute the dot product
We distribute the vector \( a \) over the expression \( (a \cdot c)b - (a \cdot b)c \):

\[
a \cdot \left[ (a \cdot c)b - (a \cdot b)c \right] = (a \cdot c)(a \cdot b) - (a \cdot b)(a \cdot c)
\]

### Step 2: Simplify the expression
The expression simplifies as follows:

\[
(a \cdot c)(a \cdot b) - (a \cdot b)(a \cdot c)
\]

Notice that both terms are identical, except for the order of multiplication, which doesn't matter because of the commutative property of scalar multiplication. Therefore:

\[
(a \cdot c)(a \cdot b) - (a \cdot b)(a \cdot c) = 0
\]

### Conclusion:
Since the dot product is zero, this means that the vector \( a \) is perpendicular to the vector \( (a \cdot c)b - (a \cdot b)c \). Therefore, we have successfully proven that the vectors are perpendicular.

---

Would you like more details on any specific part of the proof or further clarifications?

Here are some related questions for deeper understanding:
1. What are the properties of the dot product?
2. Why does a dot product of zero imply perpendicularity?
3. How can we interpret this proof geometrically?
4. What are other methods to check if two vectors are perpendicular?
5. How does this result change if the vectors are not in Euclidean space?

**Tip:** When working with vector identities, always check if a scalar factor can simplify the expressions before proceeding with the full calculation.

证明：向量 a 与 (a·c)b - (a·b)c 垂直

Learn how to prove that vector a is perpendicular to the expression (a·c)b - (a·b)c using dot product properties. This detailed solution breaks down vector algebra concepts and shows step-by-step how the dot product results in zero, implying perpendicularity. Suitable for students in Grades 10-12 studying vectors and perpendicularity.

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