For the given lines r-2-2k-k- and r-2k-2-k- which of the following is-are correct -

Given lines: L_1: r=2î+ĵ 2k̂+λ(î ĵ k̂)=a+λb and L_2: r=î ĵ 2k̂+μ( 2î ĵ+k̂)=c+μd As b is not parallel to d So, lines are either intersecting or skew. Now any par ...

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Standard XII

Mathematics

Condition for Two Lines to Be Perpendicular

For the given...

Question

For the given lines $\to r = 2^i +^j - 2^k + λ (^i -^j -^k)$ and $\to r =^i -^j - 2^k + μ (- 2^i -^j +^k),$ which of the following is/are correct ?

both lines are skew

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both lines are parallel

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both lines are intersecting

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lines intersect at point

(3, 0, - 3)

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Solution

The correct option is D lines intersect at point $(3, 0, - 3)$
Given lines:
$L_{1} : \to r = 2^i +^j - 2^k + λ (^i -^j -^k) = \to a + λ \to b$ and $L_{2} : \to r =^i -^j - 2^k + μ (- 2^i -^j +^k) = \to c + μ \to d$
As $\to b$ is not parallel to $\to d$
So, lines are either intersecting or skew.
Now any parametric point on $L_{1} = (2 + λ, 1 - λ, - 2 - λ)$ and on $L_{2} = (1 - 2 μ, - 1 - μ, - 2 + μ)$
For point of intersection : $L_{1} = L_{2}$
$\Rightarrow 2 + λ = 1 - 2 μ \dots (i) 1 - λ = - 1 - μ \dots (i i) - 2 - λ = - 2 + μ \dots (i i i)$
Solving $(i)$ and $(i i)$ we get, $λ = 1, μ = - 1$
which also satisfies equation $(i i i)$
So, point of intersection is $(3, 0, - 3)$

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Condition for Two Lines to Be Perpendicular

Standard XII Mathematics

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For the given lines →r=2^i+^j−2^k+λ(^i−^j−^k) and →r=^i−^j−2^k+μ(−2^i−^j+^k), which of the following is/are correct ?

For the given lines $\to r = 2^i +^j - 2^k + λ (^i -^j -^k)$ and $\to r =^i -^j - 2^k + μ (- 2^i -^j +^k),$ which of the following is/are correct ?