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Mathematics Question on Vector Algebra

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

A

θ=π4\frac{\pi}{4}

B

θ=π3\frac{\pi}{3}

C

θ=π2\frac{\pi}{2}

D

θ=2π3\frac{2\pi}{3}

Answer

θ=2π3\frac{2\pi}{3}

Explanation

Solution

Let a→and b→be two unit vectors and θ be the angle between them.
Then,|a⃗\vec a|=|b⃗\vec b|=1
Now,a⃗\vec a+b⃗\vec b is a unit vector if |a⃗+b⃗\vec a+\vec b|=1.
|a⃗+b⃗\vec a+\vec b|=1
⇒(a⃗+b⃗\vec a+\vec b)2=1
⇒(a⃗+b⃗\vec a+\vec b).(a⃗+b⃗\vec a+\vec b)=1
⇒a⃗.a⃗+b⃗.b⃗\vec a.\vec a+\vec b.\vec b+b⃗.a⃗+b⃗.b⃗\vec b.\vec a+\vec b.\vec b=1
⇒|∣a⃗∣2+2a⃗.b⃗+∣b⃗∣2|\vec a|^2+2\vec a.\vec b+|\vec b|^2=1
⇒1+2.1.1cosθ+1=1
⇒cosθ=−12-\frac{1}{2}
⇒ θ=2π3\frac{2\pi}{3}
Hence,a⃗+b⃗\vec a+\vec b is a unit vector if θ=2π3\frac{2\pi}{3}
The correct answer is D.