The inequality `|vec(a).vec(b)|le |vec(a)||vec(b)|` is called :

Options

A. Cauchy - Schwartz
B. Triangle Inequality
C. Rolle's Theorem
D. Lagrange's Mean Value Theorem

Answer

To solve the question regarding the inequality \( |\vec{a} \cdot \vec{b}| \leq |\vec{a}||\vec{b}| \), we need to identify the name of this inequality from the given options.

### Step-by-Step Solution:

1. **Understanding the Inequality**: The inequality \( |\vec{a} \cdot \vec{b}| \leq |\vec{a}||\vec{b}| \) is a fundamental result in vector algebra. It states that the absolute value of the dot product of two vectors is less than or equal to the product of their magnitudes.

2. **Identifying the Theorem**: This inequality is known as the **Cauchy-Schwarz Inequality**. The Cauchy-Schwarz Inequality applies to any vectors in an inner product space and is a crucial result in linear algebra and analysis.

3. **Reviewing the Options**: The options provided are:
- A) Cauchy-Schwarz
- B) Triangle Inequality
- C) Rolle's Theorem
- D) LMBT (Lagrange Mean Value Theorem)

4. **Eliminating Incorrect Options**:
- **Triangle Inequality**: This states that for any two vectors \( \vec{a} \) and \( \vec{b} \), the inequality \( |\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}| \) holds. This is not the inequality in question.
- **Rolle's Theorem**: This is a result in calculus concerning the existence of stationary points of a function. It is unrelated to vector algebra.
- **LMBT (Lagrange Mean Value Theorem)**: This theorem relates to the average rate of change of a function and is also unrelated to the vector inequality in question.

5. **Conclusion**: Since the only option that correctly identifies the inequality \( |\vec{a} \cdot \vec{b}| \leq |\vec{a}||\vec{b}| \) is the first one, we conclude that the answer is:

**Answer**: Cauchy-Schwarz Inequality.