Orthogonal Projections Onto Lines in R2: Projection Theorem and Vector Components - Prof. , Study notes of Mathematics

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Dimension and Structure

7.7 The Projection Theorem and Its Implications

November 9, 2009

Orthogonal Projections Onto Lines in R

Let a be a nonzero vector in R

2

. We would like to compute the

orthogonal projection of a vector x onto the line W = span{a}.

x

y

a

x

W = span a

x 2

x 1

Solve for k

x · a − k||a||

2 = 0 ⇒ k =

x · a

||a|| 2

Hence

x 1 = ka =

x · a

||a||^2

a.

We denote the orthogonal projection of x onto the line span{a} by

proj a x =

x · a

||a|| 2

a.

Example

Find the orthogonal projection of x = (− 3 , 2) on the line L : x + 3y = 0.

Orthogonal Projections Onto Lines Through the Origin of

R

n

Theorem 7.7.

If a is a nonzero vector in R n , then every vector x in R n can be expressed

in exactly one way as

x = x 1 + x 2

where x 1 is a scalar multiple of a and x 2 is orthogonal to a (and hence to

x 1 ). The vectors x 1 and x 2 are given by the formulas

x 1 =

x · a

||a|| 2

a and x 2 = x −

x · a

||a|| 2

a

Example

Find the vector components of x along a and orthogonal to a.

I (^) x = (2, 0 , 1), a = (1, 2 , 3).

I (^) x = (5, 0 , − 3 , 7), a = (2, 1 , − 1 , −1).

Example

Find the length of the orthogonal projection of x on a.

I (^) x = (4, − 5 , 1), a = (2, 2 , 4).

I (^) x = (5, − 3 , 7 , 1), a = (7, 1 , 0 , −1).

Hint: It is easier to use

||projax|| =

x · a

||a|| 2

a

|x · a|

||a||