Find the shortest distance from the point P 2 7 4 to a poin

Find the shortest distance from the point P = (2, -7, 4) to a point on the line given by l: (x, y, z) = (7t, 4t, 5t). The distance is

distance,s = sqrt[( 2-7t)^2 + (-7 -4t)^2 + (4-5t)^2 ]

To find the shortest distance take ds/dt = [( 2-7t)^2 + (-7 -4t)^2 + (4-5t)^2 ]^-1/2/2{ (2*-7(2-7t) +2*-4(-7-4t) +2*-5(4 -5t) }

ds/dt =0 ; 2*-7(2-7t) +2*-4(-7-4t) +2*-5(4 -5t) =0

-14(2-7t) -8(-7 -4t) -10(4 -5t) =0

-28 + 98t +56 +32t -40 +50t =0

180t - 12=0

t = 12/180 = 2/30 = 1/15

Now plug this value to find the shortest distnace

s = sqrt[( 2- 7/15)^2 +(-7 -4/15)^2 +(4-5/15)^2]

=sqrt[(23/15)^2 +(109/15)^2 + (1/15)^2]

= sqrt( 12411/225)

= 111.40/15

= 7.42 units