L1 and L2 are two lines whose vector equations are L1 - r - - cos- - -3 - - -2sin- - - cos- - -3k- L2 - r - -a- - b- - ck- where - and - are scalars and - is the acute angle between L1 and L2-If the angle - is independent of - then the value of - is

Both the lines pass through the origin. Let nbsp;line L_1 is parallel to the vector V_1. ⇒V_1 = (cosθ + √(3))î + (√(2)sinθ ) ĵ +(cosθ √(3)) k̂ and L_2 is ...

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Byju's Answer

Standard XI

Mathematics

Angle between Two Line Segments

L1 and L2 are...

Question

$L_{1}$ and $L_{2}$ are two lines whose vector equations are
$L_{1} : \to r = λ (cos θ + \sqrt{3})^i + (\sqrt{2} sin θ)^j + (cos θ - \sqrt{3})^k$
$L_{2} : \to r = μ (a^i + b^j + c^k),$ where $λ$ and $μ$ are scalars and $α$ is the acute angle between $L_{1}$ and $L_{2} .$
If the angle $α$ is independent of $θ,$ then the value of $α$ is

\frac{π}{6}

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\frac{π}{4}

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\frac{π}{3}

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\frac{π}{2}

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Solution

The correct option is A $\frac{π}{6}$
Both the lines pass through the origin.

Let line $L_{1}$ is parallel to the vector $_{1} .$
$\Rightarrow_{1} = (cos θ + \sqrt{3})^i + (\sqrt{2} sin θ)^j + (cos θ - \sqrt{3})^k$

and $L_{2}$ is parallel to the vecor $_{2}$
$\Rightarrow_{2} = a^i + b^j + c^k$
$∴ cos α = \frac{_{1} ._{2}}{∣ ∣ ∣_{1} ∣ ∣ ∣ ∣ ∣ ∣_{2} ∣ ∣ ∣}$
$= \frac{a (cos θ + \sqrt{3}) + (b \sqrt{2}) sin θ + c (cos θ - \sqrt{3})}{\sqrt{a^{2} + b^{2} + c^{2}} \sqrt{(cos θ + \sqrt{3})^{2} + 2 {sin}^{2} θ + (cos θ - \sqrt{3})^{2}}}$
$= \frac{(a + c) cos θ + b \sqrt{2} sin θ + (a - c) \sqrt{3}}{\sqrt{a^{2} + b^{2} + c^{2}} \sqrt{2 + 6}}$

For $cos α$ to be independent of $θ,$ we get $a + c = 0$ and $b = 0$
$∴ cos α = \frac{2 a \sqrt{3}}{a \sqrt{2} \times 2 \sqrt{2}} = \frac{\sqrt{3}}{2}$
$\Rightarrow α = \frac{π}{6}$

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Similar questions

L_{1}

and

L_{2}

are two lines whose vector equations are

L_{1} : \to r = λ (cos θ + \sqrt{3})^i + (\sqrt{2} sin θ)^j + (cos θ - \sqrt{3})^k

L_{2} : \to r = μ (a^i + b^j + c^k),

where

λ

and

μ

are scalars and

α

is the acute angle between

L_{1}

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If the angle

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L_{1}

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L_{2} : \to r = μ (a^i + b^j + c^k),

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α

is the acute angle between

L_{1}

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If the angle

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L_{2} : r = (2^i + 9^j + 13^k) + λ (^i + 2^j + 3^k)

and

L_{2} : r = (- 3^i + 7^j + p^k) + μ (-^i + 2^j - 3^k)

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and

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are

L_{1}

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are two lines. If the reflection of

L_{1}

L_{2}

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L_{1}

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Q. The vector equations of two lines

L_{1}

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(11, - 11, - 1)

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L1 and L2 are two lines whose vector equations are L1:→r=λ(cosθ+√3)^i+(√2sinθ)^j+(cosθ−√3)^k L2:→r=μ(a^i+b^j+c^k), where λ and μ are scalars and α is the acute angle between L1 and L2. If the angle α is independent of θ, then the value of α is