Question

For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true?

• A. \( \vec{a} \cdot \vec{b} \geq |\vec{a}| \, |\vec{b}| \)
• B. \( \vec{a} \cdot \vec{b} = |\vec{a}| \, |\vec{b}| \)
• C. \( \vec{a} \cdot \vec{b} \leq |\vec{a}| \, |\vec{b}| \)
• D. \( \vec{a} \cdot \vec{b} \geq -|\vec{a}| \, |\vec{b}| \)

Hint:

For any two vectors, the magnitude of their dot product is always less than or equal to the product of their magnitudes.

Answer 1

The Correct Option is C

Step 1: Use the dot product definition.
The dot product \( \vec{a} \cdot \vec{b} \) is given by: \[ \vec{a} \cdot \vec{b} = |\vec{a}| \, |\vec{b}| \cos \theta, \] where \( \theta \) is the angle between \( \vec{a} \) and \( \vec{b} \). Step 2: Analyze the range of \( \cos \theta \).
Since \( -1 \leq \cos \theta \leq 1 \), it follows that: \[ -|\vec{a}| \, |\vec{b}| \leq \vec{a} \cdot \vec{b} \leq |\vec{a}| \, |\vec{b}|. \] Step 3: Conclusion.
The correct answer is: \[ \boxed{\vec{a} \cdot \vec{b} \leq |\vec{a}| \, |\vec{b}|}. \]