Let overset arrow V(θ )=(cos θ + sec ⁡ θ ) hata+(cos ⁡ θ - s e c θ ) hatb , where hata and hatb are unit vectors and the angle between hata and hatb is 60o , then the minimum value of | overset arrow V|4 is equal to

Question

Let overset arrow V(θ )=(cos θ + sec ⁡ θ ) hata+(cos ⁡ θ - s e c θ ) hatb , where hata and hatb are unit vectors and the angle between hata and hatb is 60o , then the minimum value of | overset arrow V|4 is equal to (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

(| overset arrow V|)2=(|(cos θ + sec ⁡ θ ) hata + (cos ⁡ θ - sec ⁡ θ ) hatb|)2 =(cos θ + sec ⁡ θ )2+(cos ⁡ θ - sec ⁡ θ )2+(.cos ⁡ θ - sec ⁡ θ .)(.cos ⁡ θ + sec ⁡ θ .) =3cos2 θ +sec2 ⁡ θ ≥ 2 √3 ⇒ least value of | overset arrow V|2=2√3 ⇒ least value of | overset arrow V|4=12

Q. Let V→(θ)=(cosθ+sec⁡θ)a^+(cos⁡θ−secθ)b^ , where a^ and b^ are unit vectors and the angle between a^ and b^ is 60o , then the minimum value of ∣∣​V→∣∣​4 is equal to

(∣∣​V→∣∣​)2=(∣∣​(cosθ+sec⁡θ)a^+(cos⁡θ−sec⁡θ)b^∣∣​)2 =(cosθ+sec⁡θ)2+(cos⁡θ−sec⁡θ)2+(cos⁡θ−sec⁡θ)(cos⁡θ+sec⁡θ) =3cos2θ+sec2⁡θ≥23​ ⇒ least value of ∣∣​V→∣∣​2=23​ ⇒ least value of ∣∣​V→∣∣​4=12

Q. Let $V \to (θ) = (cos θ + sec θ) \overset{a}{^} + (cos θ - sec θ) \hat{b}$ , where $\overset{a}{^}$ and $\hat{b}$ are unit vectors and the angle between $\overset{a}{^}$ and $\hat{b}$ is $6 0^{o}$ , then the minimum value of $∣ ∣ V \to ∣ ∣^{4}$ is equal to