If 0° ≤
θ ≤ 90°, as in this case, the scalar projection of
a on
b coincides with the
length of the
vector projection.
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:
![{\displaystyle s=\left\|\mathbf {a} \right\|\cos \theta =\mathbf {a} \cdot \mathbf {\hat {b}} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2d84231eaf0182d9a24047f0b9caa4d66d829a7)
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
.
The term scalar component refers sometimes to scalar projection, as, in Cartesian coordinates, the components of a vector are the scalar projections in the directions of the coordinate axes.
The scalar projection is a scalar, equal to the length of the 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)
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
:
![{\displaystyle {\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|}}=\cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd5016be1ec62745c882cd98e451546934826111)
By this property, the definition of the scalar projection
becomes:
![{\displaystyle s=\left\|\mathbf {a} _{1}\right\|=\left\|\mathbf {a} \right\|\cos \theta =\left\|\mathbf {a} \right\|{\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|}}={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c793da0f1adfe0fffc0f6f8284b602a01a3841d)
Properties
The scalar projection has a negative sign if
. It coincides with the length of the corresponding vector projection if the angle is smaller than 90°. More exactly, if the vector projection is denoted
and its length
:
if
,
if
.
See also
Sources