If a unit vector veca makes angles π/3 with hati , π /4 with hatj and θ ∈ (0 , π) with hatk, then a value of θ is: -

Question

If a unit vector veca makes angles π/3 with hati , π /4 with hatj and θ ∈ (0 , π) with hatk, then a value of θ is: - (A) (5 π/12) (B) (5 π/6) (C) (2 π/3) (D) (π/4)

Tardigrade Admin · Accepted Answer

cos2 α + cos2 β + cos2 &gamma; = 1 (1/4) + (1/2) + cos2 &gamma; = 1 cos2 &gamma; = 1 - (3/4) = (1/4) cos2 &gamma; = ± (1/2) ⇒ &gamma; = (π/3) or (2π/3)

Q. If a unit vector $a$ makes angles $π /3$ with $\hat{i}, π /4$ with $\hat{j}$ and $θ \in (0, π)$ with $\hat{k}$ , then a value of $θ$ is : -

$\frac{5 π}{12}$

$\frac{5 π}{6}$

$\frac{2 π}{3}$

$\frac{π}{4}$

$cos^{2} α + cos^{2} β + cos^{2} γ = 1$
$\frac{1}{4} + \frac{1}{2} + cos^{2} γ = 1$
$cos^{2} γ = 1 - \frac{3}{4} = \frac{1}{4}$
$cos^{2} γ = \pm \frac{1}{2} \Rightarrow γ = \frac{π}{3}$ or $\frac{2 π}{3}$

Q. If a unit vector a makes angles π/3 with i^,π/4 with j^​ and θ∈(0,π) with k^, then a value of θ is : -

125π​

65π​

32π​

4π​