Question

Let \(\vec a\) anb \(\vec b\) be two unit vectors and θ is the angle between them.Then,\(\vec a+\vec b\) is a unit vector if

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

Answer 1

The Correct Option is D

Let a→and b→be two unit vectors and θ be the angle between them.
Then,|\(\vec a\)|=|\(\vec b\)|=1
Now,\(\vec a\)+\(\vec b\) is a unit vector if |\(\vec a+\vec b\)|=1.
|\(\vec a+\vec b\)|=1
⇒(\(\vec a+\vec b\))2=1
⇒(\(\vec a+\vec b\)).(\(\vec a+\vec b\))=1
⇒\(\vec a.\vec a+\vec b.\vec b\)+\(\vec b.\vec a+\vec b.\vec b\)=1
⇒|\(|\vec a|^2+2\vec a.\vec b+|\vec b|^2\)=1
⇒1+2.1.1cosθ+1=1
⇒cosθ=\(-\frac{1}{2}\)
⇒ θ=\(\frac{2\pi}{3}\)
Hence,\(\vec a+\vec b\) is a unit vector if θ=\(\frac{2\pi}{3}\)
The correct answer is D.