Question

Name of the inequality \( |\vec{a} \cdot \vec{b}| \le |\vec{a}| |\vec{b}| \) is:

• A. Cauchy-Schwarz Inequality
• B. Triangle Inequality
• C. Rolle's Inequality
• D. Lagrange's Inequality

Hint:

The Triangle Inequality refers to \( |\vec{a} + \vec{b}| \le |\vec{a}| + |\vec{b}| \). Don't mix them up!

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a fundamental theorem in linear algebra and vector calculus relating the dot product and magnitudes.
Step 2: Detailed Explanation:
We know that \( \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \).
Taking the absolute value on both sides:
\[ |\vec{a} \cdot \vec{b}| = | |\vec{a}| |\vec{b}| \cos \theta | \] \[ |\vec{a} \cdot \vec{b}| = |\vec{a}| |\vec{b}| |\cos \theta| \] Since the absolute value of cosine for any real angle \( \theta \) is always less than or equal to 1 (\( |\cos \theta| \le 1 \)):
\[ |\vec{a} \cdot \vec{b}| \le |\vec{a}| |\vec{b}| \] This specific statement is named the Cauchy-Schwarz Inequality.
Step 3: Final Answer:
The name of the inequality is Cauchy-Schwarz Inequality.