Mathematics visualization
If 0° ≤ θ ≤ 90°, as in this case, the scalar projection of a on b coincides with the length of the vector projection . Vector projection of a on b (a 1 ), and vector rejection of a from b (a 2 ).In mathematics , the scalar projection of a vector
a
{\displaystyle \mathbf {a} }
b
,
{\displaystyle \mathbf {b} ,}
scalar resolute of
a
{\displaystyle \mathbf {a} }
direction of
b
,
{\displaystyle \mathbf {b} ,}
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 }
dot product ,
b
^
{\displaystyle {\hat {\mathbf {b} }}}
unit vector in the direction of
b
,
{\displaystyle \mathbf {b} ,}
‖
a
‖
{\displaystyle \left\|\mathbf {a} \right\|}
length of
a
,
{\displaystyle \mathbf {a} ,}
θ
{\displaystyle \theta }
angle between
a
{\displaystyle \mathbf {a} }
b
{\displaystyle \mathbf {b} }
[ 1]
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} }
b
{\displaystyle \mathbf {b} }
b
{\displaystyle \mathbf {b} }
Multiplying the scalar projection of
a
{\displaystyle \mathbf {a} }
b
{\displaystyle \mathbf {b} }
b
^
{\displaystyle \mathbf {\hat {b}} }
vector projection of
a
{\displaystyle \mathbf {a} }
b
{\displaystyle \mathbf {b} }
θ [ edit ] If the angle
θ
{\displaystyle \theta }
a
{\displaystyle \mathbf {a} }
b
{\displaystyle \mathbf {b} }
a
{\displaystyle \mathbf {a} }
b
{\displaystyle \mathbf {b} }
s
=
‖
a
‖
cos
θ
.
{\displaystyle s=\left\|\mathbf {a} \right\|\cos \theta .}
s
=
‖
a
1
‖
{\displaystyle s=\left\|\mathbf {a} _{1}\right\|}
The formula above can be inverted to obtain the angle , θ .
Definition in terms of a and b [ edit ] When
θ
{\displaystyle \theta }
cosine of
θ
{\displaystyle \theta }
a
{\displaystyle \mathbf {a} }
b
,
{\displaystyle \mathbf {b} ,}
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}
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\|}}\,}
The scalar projection has a negative sign if
90
∘
<
θ
≤
180
∘
{\displaystyle 90^{\circ }<\theta \leq 180^{\circ }}
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}}
‖
a
1
‖
{\displaystyle \left\|\mathbf {a} _{1}\right\|}
s
=
‖
a
1
‖
{\displaystyle s=\left\|\mathbf {a} _{1}\right\|}
0
∘
≤
θ
≤
90
∘
,
{\displaystyle 0^{\circ }\leq \theta \leq 90^{\circ },}
s
=
−
‖
a
1
‖
{\displaystyle s=-\left\|\mathbf {a} _{1}\right\|}
90
∘
<
θ
≤
180
∘
.
{\displaystyle 90^{\circ }<\theta \leq 180^{\circ }.}
^ Strang, Gilbert (2016). Introduction to linear algebra (5th ed.). Wellesley: Cambridge press. ISBN 978-0-9802327-7-6
