If e1, e2 are two unit vectors and θ is the angle between them, then cos (θ/2) is

Question

If e1, e2 are two unit vectors and θ is the angle between them, then cos (θ/2) is (A) (1/2)|e1+e2| (B) (1/2)|e1-e2| (C) (|e1 e2|/2) (D) (|e1+e2|/2|e1||e2|)

Tardigrade Admin · Accepted Answer

(e1+e2)2=e1 ⋅ e1+2 e1 ⋅ e2+e2 ⋅ e2 ⇒ |e1+e2|=|e|2+2|e1||e2| cos θ+|e2|2 ⇒ |e1+e2|2=1+2 ⋅ 1 ⋅ 1 cos θ+1 (∵ |e1|=|e2|=1) ⇒ |e1+e2|2=2(1+ cos θ) =2(2 cos 2 (θ/2)) ⇒ |e1+e2|2=4 cos 2 (θ/2) ⇒ cos (θ/2)=(1/2)|e1+e2|

Q. If $e_{1}, e_{2}$ are two unit vectors and $θ$ is the angle between them, then $cos \frac{θ}{2}$ is

$\frac{1}{2} ∣ e_{1} + e_{2} ∣$

$\frac{1}{2} ∣ e_{1} - e_{2} ∣$

$\frac{∣ e _{1} e _{2} ∣}{2}$

$\frac{∣ e _{1} + e _{2} ∣}{2 ∣ e _{1} ∣ ∣ e _{2} ∣}$

Q. If e1​,e2​ are two unit vectors and θ is the angle between them, then cos2θ​ is

21​∣e1​+e2​∣

21​∣e1​−e2​∣

2∣e1​e2​∣​

2∣e1​∣∣e2​∣∣e1​+e2​∣​

Q. If $e_{1}, e_{2}$ are two unit vectors and $θ$ is the angle between them, then $cos \frac{θ}{2}$ is

$\frac{1}{2} ∣ e_{1} + e_{2} ∣$

$\frac{1}{2} ∣ e_{1} - e_{2} ∣$

$\frac{∣ e _{1} e _{2} ∣}{2}$

$\frac{∣ e _{1} + e _{2} ∣}{2 ∣ e _{1} ∣ ∣ e _{2} ∣}$