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 \(|\vec{a} – 2\vec{b}|\) is equal to
यदि दो सदिश \(\vec{a}\) और \(\vec{b}\) ऐसे हैं कि \(|\vec{a}| = 2\), \(|\vec{b}| = 3\) और \(\vec{a} \cdot \vec{b} = 4\) है, तो \(|\vec{a} – 2\vec{b}|\) किसके बराबर है?
Options:
- \(\sqrt{2}\)
- \(2\sqrt{2}\)
- \(2\sqrt{6}\)
- 24
- \(\sqrt{2}\)
- \(2\sqrt{2}\)
- \(2\sqrt{6}\)
- 24
Correct Answer: c)
Explanation:
\(|\vec{a}-2\vec{b}|^2 = |\vec{a}|^2 – 4\vec{a}\cdot\vec{b} + 4|\vec{b}|^2 = 2^2 – 4(4) + 4(3^2) = 4 – 16 + 36 = 24\).
\(|\vec{a}-2\vec{b}| = \sqrt{24} = 2\sqrt{6}\).
\(|\vec{a}-2\vec{b}|^2 = 4 – 16 + 36 = 24 \implies |\vec{a}-2\vec{b}| = 2\sqrt{6}\)।