A First Course in Linear Algebra, 2017a

170 R n
4.7.3 Projections
In some applications, we wish to write a vector as a sum of two related vectors. Through the concept
of projections, we can find these two vectors. First, we explore an important theorem. The result of this
theorem will provide our definition of a vector projection.
Theorem 4.37: Vector Projections
Let⃗v and ⃗u be nonzero vectors. Then there exist unique vectors⃗v || and⃗v ⊥ such that
where⃗v || is a scalar multiple of ⃗u, and⃗v ⊥ is perpendicular to ⃗u.
⃗v =⃗v || +⃗v ⊥ (4.13)
Proof. Suppose 4.13 holds and ⃗v || = k⃗u. Taking the dot product of both sides of 4.13 with ⃗u and using
⃗v ⊥ •⃗u = 0, this yields
⃗v •⃗u =(⃗v || +⃗v ⊥ ) •⃗u
= k⃗u •⃗u +⃗v ⊥ •⃗u
= k‖⃗u‖ 2
which requires k = ⃗v •⃗u/‖⃗u‖ 2 . Thus there can be no more than one vector ⃗v || .Itfollows⃗v ⊥ must equal
⃗v −⃗v || . This verifies there can be no more than one choice for both⃗v || and⃗v ⊥ and proves their uniqueness.
Now let
⃗v •⃗u
⃗v || =
‖⃗u‖ 2⃗u
and let
⃗v •⃗u
⃗v ⊥ =⃗v −⃗v || =⃗v −
‖⃗u‖ 2⃗u
Then⃗v || = k⃗u where k = ⃗v•⃗u . It only remains to verify⃗v
‖⃗u‖ 2 ⊥ •⃗u = 0. But
⃗v ⊥ •⃗u
⃗v •⃗u
= ⃗v •⃗u −
‖⃗u‖ 2⃗u •⃗u
= ⃗v •⃗u −⃗v •⃗u
= 0
♠
The vector⃗v || in Theorem 4.37 is called the projection of⃗v onto ⃗u and is denoted by
⃗v || = proj ⃗u (⃗v)
We now make a formal definition of the vector projection.