Let the foot point of the perpendicular drawn from the point P(3, -1, 11) on the straight line be L.
Hence L lies on the straight line \(\frac x2 =- \frac{y- 2}3 = \frac {z - 3}4\)
\(\therefore L (2t, 2 + 3t, 3 + 4t)\) [where t is arbitrary constant]
\(\therefore\) The direction ratios of PL are \((2t - 3, 2 + 3t + 1, 3 + 4t - 11)\) or \((2t - 3, 3t + 3, 4t - 8)\).
Again the direction ratios of the straight line \(\frac x2 =- \frac{y- 2}3 = \frac {z - 3}4\) are (2, 3, 4).
Since, PL is perpendicular on the straight line.
Then,
\((2t -3).2 + (3t +3).3 + (4t - 8).4 = 0\)
or, \(4t - 6 + 9t + 9 + 16t - 32 =0\)
or, \(29t = 29\)
\(\therefore t = 1\)
Hence, \(L(2, 5, 7)\)
\(\therefore \overline {PL} = \sqrt{(2-3)^2+ (5 + 1)^2 + (7 -11)^2}{}\)
\(= \sqrt{1 ^2 + 6^2 + 4^2}\)
\(= \sqrt{1 + 36 + 16}\)
\(= \sqrt{53}\)
Hence, the length of perpendicular is √53.
