Question:
For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true ?
For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true ?
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The Cauchy-Schwarz inequality states that \( |\vec{a} \cdot \vec{b}| \le |\vec{a}||\vec{b}| \). Since any real number is less than or equal to its absolute value (\( x \le |x| \)), it follows directly that \( \vec{a} \cdot \vec{b} \le |\vec{a}||\vec{b}| \).
Updated On: Jul 4, 2026
- \( \vec{a} \cdot \vec{b} \le |\vec{a}| |\vec{b}| \)
- \( |\vec{a} + \vec{b}| \ge |\vec{a}| + |\vec{b}| \)
- \( |\vec{a} - \vec{b}| = |\vec{a}| - |\vec{b}| \)
- \( |\vec{a} \times \vec{b}| \ge |\vec{a}| |\vec{b}| \)
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The Correct Option is A
Solution and Explanation
Concept:
This question tests fundamental vector dot product and cross product inequalities, specifically the Cauchy-Schwarz Inequality for vectors, which establishes a relation between the dot product of two vectors and the product of their magnitudes.
Step 1: Analyze the definition of the dot product.
By definition, the dot product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by: \[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \] where \( \theta \) is the angle between the two vectors \( \vec{a} \) and \( \vec{b} \), such that \( 0 \le \theta \le \pi \).
Step 2: Apply the range of the cosine function.
We know that for any real angle \( \theta \): \[ \cos \theta \le 1 \] Multiplying both sides by the non-negative scalar quantity \( |\vec{a}| |\vec{b}| \), we get: \[ |\vec{a}| |\vec{b}| \cos \theta \le |\vec{a}| |\vec{b}| \cdot 1 \] Substituting the definition of the dot product back into the inequality: \[ \vec{a} \cdot \vec{b} \le |\vec{a}| |\vec{b}| \] This statement is universally true for any pair of vectors, matching option (A).
Step 1: Analyze the definition of the dot product.
By definition, the dot product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by: \[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \] where \( \theta \) is the angle between the two vectors \( \vec{a} \) and \( \vec{b} \), such that \( 0 \le \theta \le \pi \).
Step 2: Apply the range of the cosine function.
We know that for any real angle \( \theta \): \[ \cos \theta \le 1 \] Multiplying both sides by the non-negative scalar quantity \( |\vec{a}| |\vec{b}| \), we get: \[ |\vec{a}| |\vec{b}| \cos \theta \le |\vec{a}| |\vec{b}| \cdot 1 \] Substituting the definition of the dot product back into the inequality: \[ \vec{a} \cdot \vec{b} \le |\vec{a}| |\vec{b}| \] This statement is universally true for any pair of vectors, matching option (A).
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