If an angle between the line, and the plane, x – 2y – kz = 3 is , then a value of k, is

Engineering

Mathematics

theorem in space

Distance from a Plane

Equation of Straight Line in 3D and Distance Of a Point From Line

Question

If an angle between the line, $\frac{x + 1}{2} = \frac{y - 2}{1} = \frac{z - 3}{- 2}$ and the plane, x – 2y – kz = 3 is $\cos^{- 1} (\frac{2 \sqrt{2}}{3})$ , then a value of k, is

$\frac{- 5}{3}$

$\frac{- 3}{5}$

$\sqrt{\frac{5}{3}}$

$\sqrt{\frac{3}{5}}$

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$\vec{L} ⟨ \begin{matrix} 2 & 1 & - 1 \end{matrix} ⟩ and \vec{n} ⟨ \begin{matrix} 1 & - 2 & - k \end{matrix} ⟩$

sin θ = $| \frac{2 - 2 + 2 k}{3 \sqrt{k^{2} + 5}} | = \frac{1}{3}$

⇒ | 2k | = $\sqrt{k^{2} + 5}$ ⇒ 3k² = 5 ⇒ k = ±. $\sqrt{\frac{5}{3}}$