If a unit vector a makes an angle -3 with - -4 with - and -0- - with k- then a value of - is -A- 5 -12

Unit vector a makes angles π3 with î, nbsp;π4 with ĵ and nbsp;θ with k̂. ∴cosα=12, cosβ=1√(2) cos^2α+cos^2β+cos^2θ=1 ⇒14+12+cos^2θ=1 ⇒cosθ =±12 ∴θ=π3 or 2π3 ...

If a unit vec...

Question

If a unit vector $\to a$ makes an angle $\frac{π}{3}$ with $^i$ , $\frac{π}{4}$ with $^j$ and $θ \in (0, π)$ with $^k$ , then a value of $θ$ is :

\frac{5 π}{12}

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\frac{2 π}{3}

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\frac{π}{4}

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\frac{5 π}{12}

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Solution

The correct option is C $\frac{2 π}{3}$
Unit vector $\to a$ makes an angle $\frac{π}{3}$ with $^i$ , $\frac{π}{4}$ with $^j$
$∴ cos α = \frac{1}{2}, cos β = \frac{1}{\sqrt{2}}$
${cos}^{α} + {cos}^{2} β + {cos}^{2} θ = 1$
$\Rightarrow \frac{1}{4} + \frac{1}{2} + {cos}^{2} θ = 1$
$\Rightarrow cos θ = \pm \frac{1}{2}$
$∴ θ = \frac{π}{3}$ or $\frac{π}{3}$

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