Let $L_{1} : x = y = z$
$L_{2} : x - 1 = y - 2 = z - 3$ be two lines.
Let from origin O(0, 0, 0) on $L_{1}$ , perpendicular is drawn to $L_{2}$ has foot A. Segment OA is rotated about O by an angle $90^{0}$ such that $L_{2}$ moves along with it, without changing its direction and becomes $L_{3}$ . A becomes B $(α, β, γ)$ then $α + β + γ$ is equal to ____________

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Detailed Solution

Let, $A (λ + 1, λ + 2, λ + 3)$

$∴ (λ + 1) + (λ + 2) + (λ + 3) = 0 \Rightarrow λ = - 6 \Rightarrow A (- 5, - 4, - 3)$

Let line $L_{1}$ and $L_{2}$ lies on plane P, then normal to the plane P will be $5 i + 4 j + 3 k .$
$\Rightarrow$ Eq. of plane P is
$5 x + 4 y + 3 z = 0$
$∴$ Dr’s of line through origin and $⊥$ to $L_{3}$ is
$(5 i + 4 j + 3 k) \times (i + j + k)$