If a line drawn from the point A( 1, 2, 1) is perpendicular to the line joining P(1, 4, 6) and Q(5, 4, 4) then find the co-ordinates of the foot of the perpendicular.

#### Solution

Let M be the foot of the perpendicular drawn from the point A (1, 2, 1) to the line joining P (1, 4, 6) and Q (5, 4, 4) .

Equation of a line passing through the points (x_{1},y_{1},z_{1}) and (x_{2},y_{2},z_{2}) is

`(x-x_1)/(x_2-x_1) = (y-y_1)/(y_2-y_1)=(z-z_1)/(z_2-z1)`

Equation of the required line passing through P (1, 4, 6) and Q( 5, 4, 4) is

`(x-1)/4=(y-4)/0=(z-6)/-2=lambda`

`x=4lambda+1;y=4;z=-2lambda+6`

∴ Coordinates of M are` (4lambda+1,4,-2lambda+6)` ..........(1)

The direction ratios of AM are

`4lambda+1-1,4-2,-2lambda+6-1`

i.e `4lambda, 2,-2lambda+5`

The direction ratios of given line are 4,0,-2.

Since AM is perpendicular to the given line

`therefore 4(4lambda)+0(2)+(-2)(-2lambda+5)=0`

`therefore lambda=1/2`

Putting ` lambda=1/2` in (i) , the coordinates of M are (3,4,5 ).

Length of perpendicular from A on the given line

`AM=sqrt((3-1)^2+(4-2)^2+(5-1)^2)=sqrt(24)units.`