Determine whether the lines L-1- 3- 5-k-1- 2- 3- k- R M-1- 3- 1-k1- -2- -3- k- R are identical or parallel-

[ L_ 1 :{( 1,3,5 ) +k( 1,2,3 ) ,k∈ R } #160;; L_ 2 :{( 1,3,1 ) +k( 1, 2, 3 ) ,k∈ R } ...

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Comparing the Ratios of Coefficients of a Linear Equation

Determine whe...

Question

Determine whether the lines $L : {(1, 3, 5) + k (- 1, 2, 3), k \in R}$ & $M : {(1, 3, 1) + k (1, - 2, - 3), k \in R}$ are identical or parallel.

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¯ r = (- 1, 2, 5) + k (- 1, 2, 5), k \in R

¯ r = (- 3, 1, 5) + k (- 3, 1, 5) + k (- 3, 1, 5), k \in R

\to r = (4, - 1, 0) + k (1, 2, 3), k \in R

\to r = (1, - 1, 2) + k (2, 4, - 5), k \in R

Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes $r . (^i -^j + 2^k) = 5$ and $r . (3^i +^j + 2^k) = 6$ .

L_{1} : \to r = λ^i, λ \in R,

L_{2} : \to r =^k + μ^j, μ \in R,

L_{3} : \to r =^i +^j + ν^k, ν \in R

Q

L_{2}

P

L_{1}

R

L_{3}

P, Q

R

P_{1} : (\to r -^i) . (^i -^j -^k) = 0

P_{2} : (\to r - (2^i -^j -^k)) . (^i - 2^k) \times (2^i -^j - 3^k) = 0

L : \to r = 5^i + λ (^i -^j -^k)

P_{1}

P_{2}

L

P_{1}

P_{2}

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