Question

Let the vectors \(\vec{a}\) and   \(\vec{b}\)  be such that |\(\vec{a}\)|\(=3\) and |\(\vec{b}\)|\(=\sqrt{\frac{2}{3}}\) ,then \(\vec{a}\times\vec{b}\) is a unit vector,if the angle between \(\vec{a} \) and \(\vec{b}\) is

• A. \(\frac{\pi}{6}\)
• B. \(\frac{\pi}{4}\)
• C. \(\frac{\pi}{3}\)
• D. \(\frac{\pi}{2}\)

Answer 1

The Correct Option is B

It is given that |\(\vec{a}\)|\(=3\),and |\(\vec{b}\)|\(=\sqrt{\frac{2}{3}}\)
We know that \(\vec{a}\times \vec{b}\)\(=|\vec{a}||\vec{b}|sin\theta\hat{n}\) ,where \(\hat{n}\) is a unit vector perpendicular to both \(\vec{a}\) and \(\vec{b}\) and θ is the angle between \(\vec{a}\space and\space\vec{b}\).
Now,\(\vec{a}\times \vec{b}\) is a unit vector if |\(\vec{a}\times \vec{b}\)|\(=1\)
|\(\vec{a}\times \vec{b}\)|=1
⇒\(|\vec{a}||\vec{b}|sin\theta\hat{n}=1\)
\(⇒3×\sqrt{\frac{2}{3}}×sinθ=1\)
\(⇒sinθ=\frac{1}{\sqrt{2}}\)
\(⇒θ=\frac{\pi}{4}\)
Hence,\(\vec{a}\times \vec{b}\) is a unit vector if the angle between \(\vec{a}\space and\space \vec{b} \space is \space \frac{\pi}{4}.\)
The correct answer is B.