The length of the perpendicular from the point (1, 2, 3) to the line…

Let L be the foot of the perpendicular drawn from the point P(1, 2, 3) to the given line.

Let the coordinates of L be

$(3λ + 6, 2λ + 7, -2λ + 7)$ ........(i)

∴ Direction ratios of PL are proportional to

$3λ + 6 - 1, 2 λ + 7 - 2, - 2λ + 7 - 3$

i.e.  $ 3λ + 5, 2λ + 5, -2λ + 4 $

Direction ratios of the given line are proportional to 3, 2, -2

Since PL is perpendicular to the given line.

$∴3(3λ + 5) + 2(2λ + 5) + (-2) (-2λ + 4) = 0 ⇒ λ = - 1$

Putting λ = -1 in (i), we obtain that the coordinates of L as (3, 5, 9).

$∴ PL = \sqrt{(3-1)^2 + (5-2)^2 + (9-3)^2}= 7 units$