FCM-Home       Projection of a Vector on a Line

   Here we introduce a construction which appears in many context: how to project a vector  x  onto the line through the origin in the direction of another vector  y.

The picture above shows how the vector  x  gets projected orthogonally onto the line  . The result is the vector projL(x)  which is colinear with  y. The following proposition tells us how to compute this projection vector.

Proposition   Let  x  be an arbitrary vector of  , and let    be a line through the origin in the direction of a nonzero vector  y. Then the projection of  x  onto    is the vector
 
       

Proof   We know that the projection vector has the direction of  y, and so there is a number    with  proj(x) · y. Further, we can write  x  as

x y + u,

where  u  is perpendicular to  . Dot producting both sides of this equation with  y  yields

which implies the claim.

We sometimes write  projy(x)  for  proj(x)  and call it the projection of  x  along  y. The vector  u x – projy(x)  is called the component of x orthogonal to y.

   Exercises   on projecting a vector onto a line