- Source: Scalar projection
In mathematics, the scalar projection of a vector
a
{\displaystyle \mathbf {a} }
on (or onto) a vector
b
,
{\displaystyle \mathbf {b} ,}
also known as the scalar resolute of
a
{\displaystyle \mathbf {a} }
in the direction of
b
,
{\displaystyle \mathbf {b} ,}
is given by:
s
=
‖
a
‖
cos
θ
=
a
⋅
b
^
,
{\displaystyle s=\left\|\mathbf {a} \right\|\cos \theta =\mathbf {a} \cdot \mathbf {\hat {b}} ,}
where the operator
⋅
{\displaystyle \cdot }
denotes a dot product,
b
^
{\displaystyle {\hat {\mathbf {b} }}}
is the unit vector in the direction of
b
,
{\displaystyle \mathbf {b} ,}
‖
a
‖
{\displaystyle \left\|\mathbf {a} \right\|}
is the length of
a
,
{\displaystyle \mathbf {a} ,}
and
θ
{\displaystyle \theta }
is the angle between
a
{\displaystyle \mathbf {a} }
and
b
{\displaystyle \mathbf {b} }
.
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
a
{\displaystyle \mathbf {a} }
on
b
{\displaystyle \mathbf {b} }
, with a negative sign if the projection has an opposite direction with respect to
b
{\displaystyle \mathbf {b} }
.
Multiplying the scalar projection of
a
{\displaystyle \mathbf {a} }
on
b
{\displaystyle \mathbf {b} }
by
b
^
{\displaystyle \mathbf {\hat {b}} }
converts it into the above-mentioned orthogonal projection, also called vector projection of
a
{\displaystyle \mathbf {a} }
on
b
{\displaystyle \mathbf {b} }
.
Definition based on angle θ
If the angle
θ
{\displaystyle \theta }
between
a
{\displaystyle \mathbf {a} }
and
b
{\displaystyle \mathbf {b} }
is known, the scalar projection of
a
{\displaystyle \mathbf {a} }
on
b
{\displaystyle \mathbf {b} }
can be computed using
s
=
‖
a
‖
cos
θ
.
{\displaystyle s=\left\|\mathbf {a} \right\|\cos \theta .}
(
s
=
‖
a
1
‖
{\displaystyle s=\left\|\mathbf {a} _{1}\right\|}
in the figure)
The formula above can be inverted to obtain the angle, θ.
Definition in terms of a and b
When
θ
{\displaystyle \theta }
is not known, the cosine of
θ
{\displaystyle \theta }
can be computed in terms of
a
{\displaystyle \mathbf {a} }
and
b
,
{\displaystyle \mathbf {b} ,}
by the following property of the dot product
a
⋅
b
{\displaystyle \mathbf {a} \cdot \mathbf {b} }
:
a
⋅
b
‖
a
‖
‖
b
‖
=
cos
θ
{\displaystyle {\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|}}=\cos \theta }
By this property, the definition of the scalar projection
s
{\displaystyle s}
becomes:
s
=
‖
a
1
‖
=
‖
a
‖
cos
θ
=
‖
a
‖
a
⋅
b
‖
a
‖
‖
b
‖
=
a
⋅
b
‖
b
‖
{\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\|}}\,}
Properties
The scalar projection has a negative sign if
90
∘
<
θ
≤
180
∘
{\displaystyle 90^{\circ }<\theta \leq 180^{\circ }}
. 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
a
1
{\displaystyle \mathbf {a} _{1}}
and its length
‖
a
1
‖
{\displaystyle \left\|\mathbf {a} _{1}\right\|}
:
s
=
‖
a
1
‖
{\displaystyle s=\left\|\mathbf {a} _{1}\right\|}
if
0
∘
≤
θ
≤
90
∘
,
{\displaystyle 0^{\circ }\leq \theta \leq 90^{\circ },}
s
=
−
‖
a
1
‖
{\displaystyle s=-\left\|\mathbf {a} _{1}\right\|}
if
90
∘
<
θ
≤
180
∘
.
{\displaystyle 90^{\circ }<\theta \leq 180^{\circ }.}
See also
Scalar product
Cross product
Vector projection
Sources
Dot products - www.mit.org
Scalar projection - Flexbooks.ck12.org
Scalar Projection & Vector Projection - medium.com
Lesson Explainer: Scalar Projection | Nagwa
References
Kata Kunci Pencarian:
- Scalar projection
- Vector projection
- Dot product
- Radial velocity
- Direction (geometry)
- Projection (mathematics)
- Principal component analysis
- Vector notation
- Pi (disambiguation)
- PR
