In mathematics, the scalar projection of a
vector
on (or onto) a vector
also known as the
scalar resolute of
in the direction of
is given by:
where the operator denotes a dot product, is the unit vector in the direction of is the
length
of
and
is the
angle between
and
.
[1]
The term scalar component refers sometimes to scalar projection, as, in
coordinate axes
.
The scalar projection is a
orthogonal projection
of
on
, with a negative sign if the projection has an opposite direction with respect to
.
Multiplying the scalar projection of on by converts it into the above-mentioned orthogonal projection, also called vector projection of on .
Definition based on angle θ
If the angle between and is known, the scalar projection of on can be computed using
- ( in the figure)
The formula above can be inverted to obtain the angle, θ.
Definition in terms of a and b
When is not known, the
cosine
of
can be computed in terms of
and
by the following property of the
dot product :
By this property, the definition of the scalar projection becomes:
Properties
The scalar projection has a negative sign if . It coincides with the
if the angle is smaller than 90°. More exactly, if the vector projection is denoted
and its length
:
- if
- if
See also
Sources
References