Hi,
I hope this is relatively clear and also that other people agree with me.
The vector equation of a line is of the form r.n = a.n, where r = (x,y,z), n is some vector perpendicular to the plane, and a the coordinates of a point in the plane. the . appearing is the Scalar product defined by a.d =(A,B,C).(D,E,F) = AD + BE + CF, just to clear up notation.
The vector equation for a line is r = c + Ld, where c is some point on the line, L is any real number, and d is the direction of the line, i.e. some vector parallel to the line.
So here we know that if the coordinates of P are described by the vector r, then r.n = 6. We also know, reading off from the equation of the plane, that n = (2,1,-1). Furthermore, as P is at the foot of a perpendicular, we must have that the line joining P to (1,1,9) is parallel to (2,1,-1).
Therefore we have the following two equations for the point P described by position vector r:
r.n = 6 (1)
c + Ln = r (2)
Where we choose c = (1,1,9) and n = (2,1,-1). Now what we want to do firstly is to find the value of L. This can be done by "dotting" equation 2 with the vector n, to obtain:
c.n + L n.n = r.n
And then as equation (1) says r.n = 6, we have:
c.n + Ln.n = 6
Since c and N are known we can work out :
c.n = (1,1,9).(2,1,-1) = 2 + 1 -9 = -6 ;
n.n = (2,1,-1) = 6.
hence we have:
c.n + Ln.n = -6 + 6L = 6 => L = 2
Then we have, from equation 2:
r = c + Ln = c + 2n = (1,1,9) + 2*(2,1,-1) = (1,1,9) + (4,2,-2) = (5,3,-7).
To check that this is right take (5,3,-7).(2,1,-1) = 10 + 3 - 7 = 13 -7 = 6.
So anyway the coordinates of P are (5,3,-7).
Notation: here I have used r, n, a, etc as vectors and A, B, C as pure numbers. Usually vectors should be in bold or underlined to avoid confusion with pure numbers. Can't think how to avoid the clash so clearly in general.