If line $\frac{x + 1}{λ} = \frac{y - 1}{1} = \frac{z + 2}{- 4}$ is perpendicular to the plane $2 x + 2 y - 8 z + 5 = 0$ . Then the value of λ is _____________.

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Solution

$Given line, \frac{x + 1}{λ} = \frac{y - 1}{1} = \frac{z + 2}{- 4} is perpendicular to the plane 2 x + 2 y - 8 z + 5 = 0$
$Normal \vec{n} for the plane is \vec{n} = 2 \hat{i} + 2 \hat{j} - 8 \hat{k} and direction vector for line \vec{b} = λ \hat{i} + \hat{j} - 4 \hat{k}$

Since line is perpendicular to plane
$i . e . \frac{2}{λ} = \frac{2}{1} = \frac{- 8}{- 4} i . e . \vec{n} and \vec{b} have proportional vectors i . e . \frac{2}{λ} = \frac{2}{1} i . e . λ = 1$

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