If a unit vector $\vec{r}$ makes angles $\frac{\pi}{3}$ with $\hat{i}$, $\frac{\pi}{4}$ with $\hat{j}$ and $\theta \in(0, \pi)$ with $\hat{k}$, then a value of $\theta$ is

JEE Main · 2019 · 09 Apr 2019 · Shift 2 · Vector Algebra

If a unit vector

r

makes angles

\frac{π}{3}

with

\hat{i}

\frac{π}{4}

with

\hat{j}

and

θ \in (0, π)

with

\hat{k}

, then a value of

θ

Answer: (C)

For a unit vector the squares of its direction cosines sum to

1

cos^{2} α + cos^{2} β + cos^{2} γ = 1

. With

α = \frac{π}{3}

and

β = \frac{π}{4}

(\frac{1}{2})^{2} + (\frac{1}{2})^{2} + cos^{2} θ = 1

, so

\frac{1}{4} + \frac{1}{2} + cos^{2} θ = 1

and

cos^{2} θ = \frac{1}{4}

, giving

cos θ = \pm \frac{1}{2}

. For

θ \in (0, π)

the value that matches the choices is

θ = \frac{2 π}{3}

JEE Main · 2019 · 09 Apr 2019 · Shift 2 · Single correct option