Projections, Distances

Projection

Definition

The projection of onto :

Definition

The projection of perpendicular po :

Projection.png

Distances from Points to Lines and Planes

Shortest Distance From a Point to a Line

Find the shortest distance from the point to the line which passes through the point (notes) with the direction vector

Point to Line Distance.png

Thus the shortest distance from to is

Find the point on that is closest to :

So

Shortest Distance From a Point to a plane

Find the shortest distance from the point to the plane with equation .

Point to Plane Distance.png

We see that lies on because . From the scalar equation, we also have

Thus the shortest distance from to is

Find the point on that is closest to .

So