Question

Question #9a746

Answer 1

`(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)`