vector projection

vector projection

The principle used in the

projection of line segment

a line, which results a

line segment

, may be extended to concern the

projection

of a vector

u

→

on another non-zero vector

v

→

, resulting a vector.

This projection vector, the so-called

vector projection

u

→

v

→

will be

http://planetmath.org/node/6178

parallel

to

v

→

.  It could have the length (

http://planetmath.org/Vector

) equal to

|

u

→

|

multiplied by the cosine of the

inclination

angle between the lines of

u

→

and

v

→

, as in the case of line segment.

But better than that “inclination angle” is to take the

http://planetmath.org/node/6178

angle between the both vectors

u

→

and

v

→

which may also be obtuse or

straight

; in these cases the cosine is

negative

which is suitable to cause the projection vector

u

→

v

→

to have the

direction

to

v

→

(

u

→

v

→

↓

↑

v

→

).  In all cases we define the

vector projection

or the

vector component

of

u

→

along

v

→

as

u

→

v

→

:=

|

u

→

|

⁢

cos

⁡

(

u

→

,

v

→

)

⁢

v

→

∘

(1)

where

v

→

∘

is the

unit vector

having the

http://planetmath.org/node/6178

same direction as

v

→

(i.e.,

v

→

∘

⇈

v

→

).  For the that if

u

→

=

0

→

and the angle is , then also the vector projection is the

zero vector

.

Using the

expression

for the

http://planetmath.org/node/6178

cosine of the angle between vectors and for the unit vector we thus have

u

→

v

→

=

|

u

→

|

⁢

u

→

⋅

v

→

|

u

→

|

⁢

|

v

→

|

⁢

v

→

|

v

→

|

.

This is to

u

→

v

→

=

u

→

⋅

v

→

|

v

→

|

⁢

|

v

→

|

⁢

v

→

,

(2)

where the denominator is the

scalar square

of

v

→

:

u

→

v

→

=

u

→

⋅

v

→

v

→

⋅

v

→

⁢

v

→

(3)

One can also write from (1) the alternative form

u

→

v

→

=

(

u

→

⋅

v

→

∘

)

⁢

v

→

∘

,

(4)

where the “coefficient”

u

→

⋅

v

→

∘

of the unit vector

v

→

∘

is called the

scalar projection

or the

scalar component

of

u

→

along

v

→

.

Remark 1.

The vector projection

u

→

v

→

of

u

→

along

v

→

is sometimes denoted by

proj

v

→

⁢

u

→

.

Remark 2.

If one subtracts (

http://planetmath.org/DifferenceOfVectors

) from

u

→

the vector component

u

→

v

→

, then one has another

component

of

u

→

such that the both components are

orthogonal

to each other (and their sum (

http://planetmath.org/SumVector

) is

u

→

); the orthogonality of the components follows from

(

u

→

-

u

→

v

→

)

⋅

u

→

v

→

=

u

→

⋅

v

→

v

→

⋅

v

→

⁢

u

→

⋅

v

→

-

(

u

→

⋅

v

→

v

→

⋅

v

→

)

2

⁢

v

→

⋅

v

→

=

0

.

Remark 3.

The usual “component form”

u

→

=

x

⁢

i

→

+

y

⁢

j

→

+

z

⁢

k

→

of vectors in the

cartesian coordinate system

of

ℝ

3

that the orthogonal (

http://planetmath.org/OrthogonalVectors

) vector components of

u

→

along the unit vectors

i

→

,

j

→

,

k

→

are

u

→

i

→

=

x

⁢

i

→

,

u

→

j

→

=

y

⁢

j

→

,

u

→

k

→

=

z

⁢

k

→

and the scalar components are

x

,

y

,

z

, respectively.

Title

vector projection

Canonical

name

VectorProjection

Date of creation

2013-03-22 19:05:40

Last modified on

2013-03-22 19:05:40

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Definition

Classification

msc 51N99

Classification

msc 51M04

Classification

msc 51F20

Related topic

Projection

Related topic

GramSchmidtOrthogonalization

Defines

vector component

Defines

scalar projection

Defines

scalar component

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