How would one prove this preposition Distance from a point t

How would one prove this preposition?

Distance from a point to a plane Proposition 5 Consider any plane P and any point t not on P (a) If q is a point on P, thenq is nearest to i if and only if (z-q)-(r-)- (b) There exists a unique point nearest to . for each f on P

Solution

a)

Given P is a plane and q is a point on P.

Let r be point on P

Then vector (r-q) is along the plane P.

we know that the shortest distance or the nearest distance from an external point to a plane is the line of perpendicular to the plane .

Hence if q is the nearest point to the plane then the vector (x-p) should be perpendicular to the vector (r-q)

hence dot product is zero.

(x-q)(r-q)=0

b)A line has only perpendicular from a point x . we cannot draw two perpendiclulars and hence we have only one nearest point to x