Question #9a746

1 Answer
Feb 25, 2018

#(3,4,5)#

Explanation:

The simplest way of solving this is by using vectors.

The vector #vec{PQ} = (5,4,4)-(1,4,6) = (4,0,-2)#.

The foot of the perpendicular #R# from #A# is a point on #bar{PQ}#, so its coordinate vector is of the form
# m(5,4,4)+(1-m)(1,4,6) = (1+4m,4,6-2m)#
so that the vector #vec{AR}# is

#vec{AR} = (1+4m,4,6-2m)-(1,2,1)=(4m,3,5-2m)#

Since #vec{AR}# is perpendicular to #vec{PQ}#, we have

# (4,0,-2) cdot (4m,3,5-2m) = 0 implies 20m-10=0 implies m=1/2 #

Thus, the coordinates of the foot of the perpendicular are
# (1+4times 1/2,4,6-2times 1/2) = (3,4,5)#