Find the coordinates of the foot of the perpendicular drawn from the point A(1,8,4) to the line joining the points B(0,-1,3) and C(2,-3,-1) The answer in the book is: (-5)/3 ,2/3 ,19/3

equation of line joining BC will be given by 

(x-0)/2 = (y+1)/-2 = (z-3)/-4 = k

Hence any point on BC is given by x = 2k; y = -2k-1; z = -4k+3

Let D be the foot of perpendicular from A to BC and its coordinates be (2k, -2k-1, -4k+3)

 

Hence, direction ratios of AD will be (2k-1, -2k-1-8, -4k+3-4) = (2k-1, -2k-9,-4k-1)

 

Since AD and BC are perpendicular and l1l2 + m1m2+n1n2 = 0for perpendicular lines

hence, 2(2k-1) + (-2)(-2k-9) + (-4)(-4k-1) = 0

4k-2+4k+18 +16k+4 =0

24k +20=0

k = -5/6

 

hence D : x = -5/3; y = 2/3; z = 19/3