Question

If two vectors \( \vec{a} \) and \( \vec{b} \) are such that \( |\vec{a}| = 2, |\vec{b}| = 3 \) and \( \vec{a} \cdot \vec{b} = 4 \), then \( |\vec{a} - \vec{b}| = \)_____

• A. \( \sqrt{5} \)
• B. \( 13 \)
• C. \( 5 \)
• D. \( \sqrt{17} \)

Hint:

Always remember vector identity: \( |\vec{a} - \vec{b}|^2 = a^2 + b^2 - 2ab\cos\theta \).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The magnitude of the difference of two vectors can be found using the identity $|\vec{a} - \vec{b}|^2 = (\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b})$.
Step 2: Formula Derivation:
$$|\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2(\vec{a} \cdot \vec{b})$$
Step 3: Explanation:
Substitute the given values: $$|\vec{a} - \vec{b}|^2 = (2)^2 + (3)^2 - 2(4)$$ $$|\vec{a} - \vec{b}|^2 = 4 + 9 - 8 = 5$$ $$|\vec{a} - \vec{b}| = \sqrt{5}$$
Step 4: Final Answer:
The magnitude is $\sqrt{5}$.