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If the plane 2x-3y-6z=13 makes an angle ^{-1}() with the x-axi

Question

If the plane $2 x - 3 y - 6 z = 13$ makes an angle $sin^{- 1} (λ)$ with the x-axis, then find the value of $λ$ .

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If the plane

2 x - 3 y - 6 z = 13

makes an angle

sin^{- 1} (λ)

with the x-axis, then find the value of

λ

11 mins ago

Discuss this question LIVE

11 mins ago

Text solutionVerified

Direction ratios of the x-axis are

1, 0, 0

and direction ratios of normal to the plane are

2, - 3, - 6

Now, the angle between the line and the plane is given by:

sin ϕ = \frac{∣ a . n ∣}{∣ a ∣∣ n ∣}

sin ϕ = \frac{∣ a _{1} . a _{2} + b _{1} . b _{2} + c _{1} . c _{2} ∣}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} . a _{2}^{2} + b _{2}^{2} + c _{2}^{2}}

Let

ϕ

be the angle between the x-axis and the given plane.

Then,

sin ϕ = \frac{∣1 \times 2 + 0 \times ( - 3 ) + 0 \times ( - 6 ) ∣}{{ 1 ^{2} + 0 ^{2} + 0 ^{2} } { 2 ^{2} + ( - 3 ) ^{2} + ( - 6 ) ^{2} }} = \frac{2}{7}

\Rightarrow ϕ = sin^{- 1} (\frac{2}{7})

Hence,

λ = \frac{2}{7}

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Question Text	If the plane $2 x - 3 y - 6 z = 13$ makes an angle $sin^{- 1} (λ)$ with the x-axis, then find the value of $λ$ .
Answer Type	Text solution:1
Upvotes	150