Question

Let and be two unit vectors and θ is the angle between them. Then is a unit vector if (A) (B) (C) (D)

Answer 1

It is given that a →  and  b → are two unit vectors and θ is the angle between them.

Since, a →  and  b → are unit vectors, so, magnitude of unit vectors is,

| a → |=1 and | b → |=1.

Now, | a → + b → | is a unit vector if,

| a → + b → |=1 | a → + b → | 2 = 1 2 ( a → + b → )⋅( a → + b → )=1 a → ⋅( a → + b → )+ b → ⋅( a → + b → )=1

Further simplify the above equation.

a → ⋅ a → + b → ⋅ a → + a → ⋅ b → + b → ⋅ b → =1 | a → | 2 + a → ⋅ b → + b → ⋅ a → + | b → | 2 =1 1+ a → ⋅ b → + a → ⋅ b → +1=1 2+2 a → ⋅ b → =1

Simplify further,

2 a → ⋅ b → =−1 a → ⋅ b → = −1 2

The formula of dot product of a →  and  b → is,

a → ⋅ b → =| a → || b → |cosθ

Substitute the value , a → ⋅ b → = −1 2 in the above formula.

a → ⋅ b → =1×1cosθ −1 2 =cosθ θ= cos −1 ( −1 2 ) θ=120° or  2π 3  radian

Thus, the correct answer is option (D).