<s>
The	O
vector	B-Algorithm
projection	I-Algorithm
of	O
a	O
vector	O
on	O
(	O
or	O
onto	O
)	O
a	O
nonzero	O
vector	O
,	O
sometimes	O
denoted	O
(	O
also	O
known	O
as	O
the	O
vector	O
component	O
or	O
vector	O
resolution	O
of	O
in	O
the	O
direction	O
of	O
)	O
,	O
is	O
the	O
orthogonal	O
projection	B-Algorithm
of	O
onto	O
a	O
straight	O
line	O
parallel	O
to	O
.	O
</s>
<s>
where	O
is	O
a	O
scalar	O
,	O
called	O
the	O
scalar	O
projection	B-Algorithm
of	O
onto	O
,	O
and	O
is	O
the	O
unit	O
vector	O
in	O
the	O
direction	O
of	O
.	O
</s>
<s>
In	O
turn	O
,	O
the	O
scalar	O
projection	B-Algorithm
is	O
defined	O
as	O
:	O
</s>
<s>
The	O
scalar	O
projection	B-Algorithm
is	O
equal	O
to	O
the	O
length	O
of	O
the	O
vector	B-Algorithm
projection	I-Algorithm
,	O
with	O
a	O
minus	O
sign	O
if	O
the	O
direction	O
of	O
the	O
projection	B-Algorithm
is	O
opposite	O
to	O
the	O
direction	O
of	O
.	O
</s>
<s>
The	O
vector	O
component	O
or	O
vector	B-Algorithm
resolute	I-Algorithm
of	O
perpendicular	O
to	O
,	O
sometimes	O
also	O
called	O
the	O
vector	O
rejection	O
of	O
from	O
(	O
denoted	O
)	O
,	O
is	O
the	O
orthogonal	O
projection	B-Algorithm
of	O
onto	O
the	O
plane	O
(	O
or	O
,	O
in	O
general	O
,	O
hyperplane	O
)	O
orthogonal	O
to	O
.	O
</s>
<s>
Both	O
the	O
projection	B-Algorithm
and	O
rejection	O
of	O
a	O
vector	O
are	O
vectors	O
,	O
and	O
their	O
sum	O
is	O
equal	O
to	O
,	O
which	O
implies	O
that	O
the	O
rejection	O
is	O
given	O
by	O
:	O
</s>
<s>
Typically	O
,	O
a	O
vector	B-Algorithm
projection	I-Algorithm
is	O
denoted	O
in	O
a	O
bold	O
font	O
(	O
e.g.	O
</s>
<s>
)	O
,	O
and	O
the	O
corresponding	O
scalar	O
projection	B-Algorithm
with	O
normal	O
font	O
(	O
e.g.	O
</s>
<s>
In	O
some	O
cases	O
,	O
especially	O
in	O
handwriting	O
,	O
the	O
vector	B-Algorithm
projection	I-Algorithm
is	O
also	O
denoted	O
using	O
a	O
diacritic	O
above	O
or	O
below	O
the	O
letter	O
(	O
e.g.	O
,	O
or	O
a1	O
)	O
.	O
</s>
<s>
The	O
vector	B-Algorithm
projection	I-Algorithm
of	O
on	O
and	O
the	O
corresponding	O
rejection	O
are	O
sometimes	O
denoted	O
by	O
and	O
,	O
respectively	O
.	O
</s>
<s>
A	O
scalar	O
projection	B-Algorithm
can	O
be	O
used	O
as	O
a	O
scale	B-Algorithm
factor	I-Algorithm
to	O
compute	O
the	O
corresponding	O
vector	B-Algorithm
projection	I-Algorithm
.	O
</s>
<s>
The	O
vector	B-Algorithm
projection	I-Algorithm
of	O
on	O
is	O
a	O
vector	O
whose	O
magnitude	O
is	O
the	O
scalar	O
projection	B-Algorithm
of	O
on	O
with	O
the	O
same	O
direction	O
as	O
.	O
</s>
<s>
where	O
is	O
the	O
corresponding	O
scalar	O
projection	B-Algorithm
,	O
as	O
defined	O
above	O
,	O
and	O
is	O
the	O
unit	O
vector	O
with	O
the	O
same	O
direction	O
as	O
:	O
</s>
<s>
By	O
the	O
above-mentioned	O
property	O
of	O
the	O
dot	O
product	O
,	O
the	O
definition	O
of	O
the	O
scalar	O
projection	B-Algorithm
becomes	O
:	O
</s>
<s>
Similarly	O
,	O
the	O
definition	O
of	O
the	O
vector	B-Algorithm
projection	I-Algorithm
of	O
onto	O
becomes	O
:	O
</s>
<s>
In	O
two	O
dimensions	O
,	O
the	O
scalar	O
rejection	O
is	O
equivalent	O
to	O
the	O
projection	B-Algorithm
of	O
onto	O
,	O
which	O
is	O
rotated	O
90°	O
to	O
the	O
left	O
.	O
</s>
<s>
The	O
scalar	O
projection	B-Algorithm
on	O
is	O
a	O
scalar	O
which	O
has	O
a	O
negative	O
sign	O
if	O
90	O
degrees	O
<	O
θ	O
≤	O
180	O
degrees	O
.	O
</s>
<s>
It	O
coincides	O
with	O
the	O
length	O
of	O
the	O
vector	B-Algorithm
projection	I-Algorithm
if	O
the	O
angle	O
is	O
smaller	O
than	O
90°	O
.	O
</s>
<s>
The	O
vector	B-Algorithm
projection	I-Algorithm
of	O
on	O
is	O
a	O
vector	O
which	O
is	O
either	O
null	O
or	O
parallel	O
to	O
.	O
</s>
<s>
The	O
orthogonal	O
projection	B-Algorithm
can	O
be	O
represented	O
by	O
a	O
projection	B-Algorithm
matrix	O
.	O
</s>
<s>
To	O
project	O
a	O
vector	O
onto	O
the	O
unit	O
vector	O
,	O
it	O
would	O
need	O
to	O
be	O
multiplied	O
with	O
this	O
projection	B-Algorithm
matrix	O
:	O
</s>
<s>
The	O
vector	B-Algorithm
projection	I-Algorithm
is	O
an	O
important	O
operation	O
in	O
the	O
Gram	B-Algorithm
–	I-Algorithm
Schmidt	I-Algorithm
orthonormalization	B-Algorithm
of	O
vector	O
space	O
bases	O
.	O
</s>
<s>
Since	O
the	O
notions	O
of	O
vector	O
length	O
and	O
angle	O
between	O
vectors	O
can	O
be	O
generalized	O
to	O
any	O
n-dimensional	O
inner	O
product	O
space	O
,	O
this	O
is	O
also	O
true	O
for	O
the	O
notions	O
of	O
orthogonal	O
projection	B-Algorithm
of	O
a	O
vector	O
,	O
projection	B-Algorithm
of	O
a	O
vector	O
onto	O
another	O
,	O
and	O
rejection	O
of	O
a	O
vector	O
from	O
another	O
.	O
</s>
<s>
Whenever	O
they	O
do	O
n't	O
coincide	O
,	O
the	O
inner	O
product	O
is	O
used	O
instead	O
of	O
the	O
dot	O
product	O
in	O
the	O
formal	O
definitions	O
of	O
projection	B-Algorithm
and	I-Algorithm
rejection	I-Algorithm
.	O
</s>
<s>
For	O
a	O
three-dimensional	O
inner	O
product	O
space	O
,	O
the	O
notions	O
of	O
projection	B-Algorithm
of	O
a	O
vector	O
onto	O
another	O
and	O
rejection	O
of	O
a	O
vector	O
from	O
another	O
can	O
be	O
generalized	O
to	O
the	O
notions	O
of	O
projection	B-Algorithm
of	O
a	O
vector	O
onto	O
a	O
plane	O
,	O
and	O
rejection	O
of	O
a	O
vector	O
from	O
a	O
plane	O
.	O
</s>
<s>
The	O
projection	B-Algorithm
of	O
a	O
vector	O
on	O
a	O
plane	O
is	O
its	O
orthogonal	O
projection	B-Algorithm
on	O
that	O
plane	O
.	O
</s>
<s>
The	O
rejection	O
of	O
a	O
vector	O
from	O
a	O
plane	O
is	O
its	O
orthogonal	O
projection	B-Algorithm
on	O
a	O
straight	O
line	O
which	O
is	O
orthogonal	O
to	O
that	O
plane	O
.	O
</s>
<s>
For	O
a	O
given	O
vector	O
and	O
plane	O
,	O
the	O
sum	O
of	O
projection	B-Algorithm
and	I-Algorithm
rejection	I-Algorithm
is	O
equal	O
to	O
the	O
original	O
vector	O
.	O
</s>
<s>
Similarly	O
,	O
for	O
inner	O
product	O
spaces	O
with	O
more	O
than	O
three	O
dimensions	O
,	O
the	O
notions	O
of	O
projection	B-Algorithm
onto	O
a	O
vector	O
and	O
rejection	O
from	O
a	O
vector	O
can	O
be	O
generalized	O
to	O
the	O
notions	O
of	O
projection	B-Algorithm
onto	O
a	O
hyperplane	O
,	O
and	O
rejection	O
from	O
a	O
hyperplane	O
.	O
</s>
<s>
In	O
geometric	O
algebra	O
,	O
they	O
can	O
be	O
further	O
generalized	O
to	O
the	O
notions	O
of	O
projection	B-Algorithm
and	I-Algorithm
rejection	I-Algorithm
of	O
a	O
general	O
multivector	O
onto/from	O
any	O
invertible	O
k-blade	O
.	O
</s>
