To find the length of the perpendicular from point A to the line l, we first need to find the vector AP where P is a point on the line l.
The line l is given by r = 4i - 9j + 9k + \(\lambda (-2i + j - 2k)\).
Let P be a point on the line, then the position vector of P is r = (4 - 2\(\lambda\))i + (-9 + \(\lambda\))j + (9 - 2\(\lambda\))k.
The position vector of A is 3i + 8j + 5k.
Thus, AP = ((4 - 2\(\lambda\)) - 3)i + ((-9 + \(\lambda\)) - 8)j + ((9 - 2\(\lambda\)) - 5)k.
Simplifying, AP = (1 - 2\(\lambda\))i + (-17 + \(\lambda\))j + (4 - 2\(\lambda\))k.
To find the perpendicular distance, we need to minimize the magnitude of AP.
The magnitude of AP is \(\sqrt{(1 - 2\lambda)^2 + (-17 + \lambda)^2 + (4 - 2\lambda)^2}\).
Expanding and simplifying, we get:
\((1 - 2\lambda)^2 = 1 - 4\lambda + 4\lambda^2\)
\((-17 + \lambda)^2 = 289 - 34\lambda + \lambda^2\)
\((4 - 2\lambda)^2 = 16 - 16\lambda + 4\lambda^2\)
Adding these, we get:
\(5\lambda^2 - 54\lambda + 306\)
To minimize, set the derivative with respect to \(\lambda\) to zero:
\(10\lambda - 54 = 0\)
\(\lambda = 5.4\)
Substitute \(\lambda = 5.4\) back into the expression for the magnitude:
\(\sqrt{5(5.4)^2 - 54(5.4) + 306} = 15\)
Thus, the length of the perpendicular from A to l is 15.
