Formulas about vectors in three-dimensional Euclidean space
The following are important identities in vector algebra . Identities that only involve the magnitude of a vector
‖
A
‖
{\displaystyle \|\mathbf {A} \|}
dot product (scalar product) of two vectors A ·B , apply to vectors in any dimension, while identities that use the cross product (vector product) A ×B only apply in three dimensions, since the cross product is only defined there.[nb 1] [1] Josiah Willard Gibbs , if not earlier.[2]
The magnitude of a vector A can be expressed using the dot product:
‖
A
‖
2
=
A
⋅
A
{\displaystyle \|\mathbf {A} \|^{2}=\mathbf {A\cdot A} }
In three-dimensional Euclidean space , the magnitude of a vector is determined from its three components using Pythagoras' theorem :
‖
A
‖
2
=
A
1
2
+
A
2
2
+
A
3
2
{\displaystyle \|\mathbf {A} \|^{2}=A_{1}^{2}+A_{2}^{2}+A_{3}^{2}}
The Cauchy–Schwarz inequality :
A
⋅
B
≤
‖
A
‖
‖
B
‖
{\displaystyle \mathbf {A} \cdot \mathbf {B} \leq \left\|\mathbf {A} \right\|\left\|\mathbf {B} \right\|}
The triangle inequality :
‖
A
+
B
‖
≤
‖
A
‖
+
‖
B
‖
{\displaystyle \|\mathbf {A+B} \|\leq \|\mathbf {A} \|+\|\mathbf {B} \|}
The reverse triangle inequality :
‖
A
−
B
‖
≥
|
‖
A
‖
−
‖
B
‖
|
{\displaystyle \|\mathbf {A-B} \|\geq {\Bigl |}\|\mathbf {A} \|-\|\mathbf {B} \|{\Bigr |}}
The vector product and the scalar product of two vectors define the angle between them, say θ :[1] [3]
sin
θ
=
‖
A
×
B
‖
‖
A
‖
‖
B
‖
(
−
π
<
θ
≤
π
)
{\displaystyle \sin \theta ={\frac {\|\mathbf {A} \times \mathbf {B} \|}{\left\|\mathbf {A} \right\|\left\|\mathbf {B} \right\|}}\quad (-\pi <\theta \leq \pi )}
To satisfy the right-hand rule , for positive θ , vector B is counter-clockwise from A , and for negative θ it is clockwise.
cos
θ
=
A
⋅
B
‖
A
‖
‖
B
‖
(
−
π
<
θ
≤
π
)
{\displaystyle \cos \theta ={\frac {\mathbf {A} \cdot \mathbf {B} }{\left\|\mathbf {A} \right\|\left\|\mathbf {B} \right\|}}\quad (-\pi <\theta \leq \pi )}
The Pythagorean trigonometric identity then provides:
‖
A
×
B
‖
2
+
(
A
⋅
B
)
2
=
‖
A
‖
2
‖
B
‖
2
{\displaystyle \left\|\mathbf {A\times B} \right\|^{2}+(\mathbf {A} \cdot \mathbf {B} )^{2}=\left\|\mathbf {A} \right\|^{2}\left\|\mathbf {B} \right\|^{2}}
If a vector A = (Ax , Ay , Az ) makes angles α , β , γ with an orthogonal set of x- , y- and z- axes, then:
cos
α
=
A
x
A
x
2
+
A
y
2
+
A
z
2
=
A
x
‖
A
‖
,
{\displaystyle \cos \alpha ={\frac {A_{x}}{\sqrt {A_{x}^{2}+A_{y}^{2}+A_{z}^{2}}}}={\frac {A_{x}}{\|\mathbf {A} \|}}\ ,}
and analogously for angles β, γ. Consequently:
A
=
‖
A
‖
(
cos
α
i
^
+
cos
β
j
^
+
cos
γ
k
^
)
,
{\displaystyle \mathbf {A} =\left\|\mathbf {A} \right\|\left(\cos \alpha \ {\hat {\mathbf {i} }}+\cos \beta \ {\hat {\mathbf {j} }}+\cos \gamma \ {\hat {\mathbf {k} }}\right),}
with
i
^
,
j
^
,
k
^
{\displaystyle {\hat {\mathbf {i} }},\ {\hat {\mathbf {j} }},\ {\hat {\mathbf {k} }}}
The area Σ of a parallelogram with sides A and B containing the angle θ is:
Σ
=
A
B
sin
θ
,
{\displaystyle \Sigma =AB\sin \theta ,}
which will be recognized as the magnitude of the vector cross product of the vectors A and B lying along the sides of the parallelogram. That is:
Σ
=
‖
A
×
B
‖
=
‖
A
‖
2
‖
B
‖
2
−
(
A
⋅
B
)
2
.
{\displaystyle \Sigma =\left\|\mathbf {A} \times \mathbf {B} \right\|={\sqrt {\left\|\mathbf {A} \right\|^{2}\left\|\mathbf {B} \right\|^{2}-\left(\mathbf {A} \cdot \mathbf {B} \right)^{2}}}\ .}
(If A , B are two-dimensional vectors, this is equal to the determinant of the 2 × 2 matrix with rows A , B .) The square of this expression is:[4]
Σ
2
=
(
A
⋅
A
)
(
B
⋅
B
)
−
(
A
⋅
B
)
(
B
⋅
A
)
=
Γ
(
A
,
B
)
,
{\displaystyle \Sigma ^{2}=(\mathbf {A\cdot A} )(\mathbf {B\cdot B} )-(\mathbf {A\cdot B} )(\mathbf {B\cdot A} )=\Gamma (\mathbf {A} ,\ \mathbf {B} )\ ,}
where Γ(A , B ) is the Gram determinant of A and B defined by:
Γ
(
A
,
B
)
=
|
A
⋅
A
A
⋅
B
B
⋅
A
B
⋅
B
|
.
{\displaystyle \Gamma (\mathbf {A} ,\ \mathbf {B} )={\begin{vmatrix}\mathbf {A\cdot A} &\mathbf {A\cdot B} \\\mathbf {B\cdot A} &\mathbf {B\cdot B} \end{vmatrix}}\ .}
In a similar fashion, the squared volume V of a parallelepiped spanned by the three vectors A , B , C is given by the Gram determinant of the three vectors:[4]
V
2
=
Γ
(
A
,
B
,
C
)
=
|
A
⋅
A
A
⋅
B
A
⋅
C
B
⋅
A
B
⋅
B
B
⋅
C
C
⋅
A
C
⋅
B
C
⋅
C
|
,
{\displaystyle V^{2}=\Gamma (\mathbf {A} ,\ \mathbf {B} ,\ \mathbf {C} )={\begin{vmatrix}\mathbf {A\cdot A} &\mathbf {A\cdot B} &\mathbf {A\cdot C} \\\mathbf {B\cdot A} &\mathbf {B\cdot B} &\mathbf {B\cdot C} \\\mathbf {C\cdot A} &\mathbf {C\cdot B} &\mathbf {C\cdot C} \end{vmatrix}}\ ,}
Since A , B, C are three-dimensional vectors, this is equal to the square of the scalar triple product
det
[
A
,
B
,
C
]
=
|
A
,
B
,
C
|
{\displaystyle \det[\mathbf {A} ,\mathbf {B} ,\mathbf {C} ]=|\mathbf {A} ,\mathbf {B} ,\mathbf {C} |}
This process can be extended to n -dimensions.
Addition and multiplication of vectors [ edit ] Commutativity of addition:
A
+
B
=
B
+
A
{\displaystyle \mathbf {A} +\mathbf {B} =\mathbf {B} +\mathbf {A} }
Commutativity of scalar product:
A
⋅
B
=
B
⋅
A
{\displaystyle \mathbf {A} \cdot \mathbf {B} =\mathbf {B} \cdot \mathbf {A} }
Anticommutativity of cross product:
A
×
B
=
−
(
B
×
A
)
{\displaystyle \mathbf {A} \times \mathbf {B} =\mathbf {-} (\mathbf {B} \times \mathbf {A} )}
Distributivity of multiplication by a scalar over addition:
c
(
A
+
B
)
=
c
A
+
c
B
{\displaystyle c(\mathbf {A} +\mathbf {B} )=c\mathbf {A} +c\mathbf {B} }
Distributivity of scalar product over addition:
(
A
+
B
)
⋅
C
=
A
⋅
C
+
B
⋅
C
{\displaystyle \left(\mathbf {A} +\mathbf {B} \right)\cdot \mathbf {C} =\mathbf {A} \cdot \mathbf {C} +\mathbf {B} \cdot \mathbf {C} }
Distributivity of vector product over addition:
(
A
+
B
)
×
C
=
A
×
C
+
B
×
C
{\displaystyle (\mathbf {A} +\mathbf {B} )\times \mathbf {C} =\mathbf {A} \times \mathbf {C} +\mathbf {B} \times \mathbf {C} }
Scalar triple product :
A
⋅
(
B
×
C
)
=
B
⋅
(
C
×
A
)
=
C
⋅
(
A
×
B
)
=
|
A
B
C
|
=
|
A
x
B
x
C
x
A
y
B
y
C
y
A
z
B
z
C
z
|
.
{\displaystyle \mathbf {A} \cdot (\mathbf {B} \times \mathbf {C} )=\mathbf {B} \cdot (\mathbf {C} \times \mathbf {A} )=\mathbf {C} \cdot (\mathbf {A} \times \mathbf {B} )=|\mathbf {A} \,\mathbf {B} \,\mathbf {C} |={\begin{vmatrix}A_{x}&B_{x}&C_{x}\\A_{y}&B_{y}&C_{y}\\A_{z}&B_{z}&C_{z}\end{vmatrix}}.}
Vector triple product :
A
×
(
B
×
C
)
=
(
A
⋅
C
)
B
−
(
A
⋅
B
)
C
{\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )=(\mathbf {A} \cdot \mathbf {C} )\mathbf {B} -(\mathbf {A} \cdot \mathbf {B} )\mathbf {C} }
Jacobi identity :
A
×
(
B
×
C
)
+
C
×
(
A
×
B
)
+
B
×
(
C
×
A
)
=
0
.
{\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )+\mathbf {C} \times (\mathbf {A} \times \mathbf {B} )+\mathbf {B} \times (\mathbf {C} \times \mathbf {A} )=\mathbf {0} .}
Lagrange's identity :
|
A
×
B
|
2
=
(
A
⋅
A
)
(
B
⋅
B
)
−
(
A
⋅
B
)
2
{\displaystyle |\mathbf {A} \times \mathbf {B} |^{2}=(\mathbf {A} \cdot \mathbf {A} )(\mathbf {B} \cdot \mathbf {B} )-(\mathbf {A} \cdot \mathbf {B} )^{2}}
In mathematics , the quadruple product is a product of four vectors in three-dimensional Euclidean space . The name "quadruple product" is used for two different products,[5] scalar quadruple product and the vector-valued vector quadruple product or vector product of four vectors .
Scalar quadruple product [ edit ] The scalar quadruple product is defined as the dot product of two cross products :
(
a
×
b
)
⋅
(
c
×
d
)
,
{\displaystyle (\mathbf {a\times b} )\cdot (\mathbf {c} \times \mathbf {d} )\ ,}
where a, b, c, d are vectors in three-dimensional Euclidean space.[6] Binet-Cauchy identity :[6]
(
a
×
b
)
⋅
(
c
×
d
)
=
(
a
⋅
c
)
(
b
⋅
d
)
−
(
a
⋅
d
)
(
b
⋅
c
)
.
{\displaystyle (\mathbf {a\times b} )\cdot (\mathbf {c} \times \mathbf {d} )=(\mathbf {a\cdot c} )(\mathbf {b\cdot d} )-(\mathbf {a\cdot d} )(\mathbf {b\cdot c} )\ .}
or using the determinant :
(
a
×
b
)
⋅
(
c
×
d
)
=
|
a
⋅
c
a
⋅
d
b
⋅
c
b
⋅
d
|
.
{\displaystyle (\mathbf {a\times b} )\cdot (\mathbf {c} \times \mathbf {d} )={\begin{vmatrix}\mathbf {a\cdot c} &\mathbf {a\cdot d} \\\mathbf {b\cdot c} &\mathbf {b\cdot d} \end{vmatrix}}\ .}
Vector quadruple product [ edit ] The vector quadruple product is defined as the cross product of two cross products:
(
a
×
b
)
×
(
c
×
d
)
,
{\displaystyle (\mathbf {a\times b} )\mathbf {\times } (\mathbf {c} \times \mathbf {d} )\ ,}
where a, b, c, d are vectors in three-dimensional Euclidean space.[2] [7]
(
a
×
b
)
×
(
c
×
d
)
=
[
a
,
b
,
d
]
c
−
[
a
,
b
,
c
]
d
,
{\displaystyle (\mathbf {a\times b} )\mathbf {\times } (\mathbf {c} \times \mathbf {d} )=[\mathbf {a,\ b,\ d} ]\mathbf {c} -[\mathbf {a,\ b,\ c} ]\mathbf {d} \ ,}
using the notation for the triple product :
[
a
,
b
,
c
]
=
a
⋅
(
b
×
c
)
.
{\displaystyle [\mathbf {a,\ b,\ c} ]=\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\ .}
Equivalent forms can be obtained using the identity:[8] [9] [10]
[
b
,
c
,
d
]
a
−
[
c
,
d
,
a
]
b
+
[
d
,
a
,
b
]
c
−
[
a
,
b
,
c
]
d
=
0
.
{\displaystyle [\mathbf {b,\ c,\ d} ]\mathbf {a} -[\mathbf {c,\ d,\ a} ]\mathbf {b} +[\mathbf {d,\ a,\ b} ]\mathbf {c} -[\mathbf {a,\ b,\ c} ]\mathbf {d} =0\ .}
This identity can also be written using tensor notation and the Einstein summation convention as follows:
(
a
×
b
)
×
(
c
×
d
)
=
ε
i
j
k
a
i
c
j
d
k
b
l
−
ε
i
j
k
b
i
c
j
d
k
a
l
=
ε
i
j
k
a
i
b
j
d
k
c
l
−
ε
i
j
k
a
i
b
j
c
k
d
l
{\displaystyle (\mathbf {a\times b} )\mathbf {\times } (\mathbf {c} \times \mathbf {d} )=\varepsilon _{ijk}a^{i}c^{j}d^{k}b^{l}-\varepsilon _{ijk}b^{i}c^{j}d^{k}a^{l}=\varepsilon _{ijk}a^{i}b^{j}d^{k}c^{l}-\varepsilon _{ijk}a^{i}b^{j}c^{k}d^{l}}
where εijk Levi-Civita symbol .
Related relationships:
A consequence of the previous equation:[11]
|
A
B
C
|
D
=
(
A
⋅
D
)
(
B
×
C
)
+
(
B
⋅
D
)
(
C
×
A
)
+
(
C
⋅
D
)
(
A
×
B
)
.
{\displaystyle |\mathbf {A} \,\mathbf {B} \,\mathbf {C} |\,\mathbf {D} =(\mathbf {A} \cdot \mathbf {D} )\left(\mathbf {B} \times \mathbf {C} \right)+\left(\mathbf {B} \cdot \mathbf {D} \right)\left(\mathbf {C} \times \mathbf {A} \right)+\left(\mathbf {C} \cdot \mathbf {D} \right)\left(\mathbf {A} \times \mathbf {B} \right).}
In 3 dimensions, a vector D can be expressed in terms of basis vectors {A ,B ,C } as:[12]
D
=
D
⋅
(
B
×
C
)
|
A
B
C
|
A
+
D
⋅
(
C
×
A
)
|
A
B
C
|
B
+
D
⋅
(
A
×
B
)
|
A
B
C
|
C
.
{\displaystyle \mathbf {D} \ =\ {\frac {\mathbf {D} \cdot (\mathbf {B} \times \mathbf {C} )}{|\mathbf {A} \,\mathbf {B} \,\mathbf {C} |}}\ \mathbf {A} +{\frac {\mathbf {D} \cdot (\mathbf {C} \times \mathbf {A} )}{|\mathbf {A} \,\mathbf {B} \,\mathbf {C} |}}\ \mathbf {B} +{\frac {\mathbf {D} \cdot (\mathbf {A} \times \mathbf {B} )}{|\mathbf {A} \,\mathbf {B} \,\mathbf {C} |}}\ \mathbf {C} .}
These relations are useful for deriving various formulas in spherical and Euclidean geometry. For example, if four points are chosen on the unit sphere, A, B, C, D , and unit vectors drawn from the center of the sphere to the four points, a, b, c, d respectively, the identity:
(
a
×
b
)
⋅
(
c
×
d
)
=
(
a
⋅
c
)
(
b
⋅
d
)
−
(
a
⋅
d
)
(
b
⋅
c
)
,
{\displaystyle (\mathbf {a\times b} )\mathbf {\cdot } (\mathbf {c\times d} )=(\mathbf {a\cdot c} )(\mathbf {b\cdot d} )-(\mathbf {a\cdot d} )(\mathbf {b\cdot c} )\ ,}
in conjunction with the relation for the magnitude of the cross product:
‖
a
×
b
‖
=
a
b
sin
θ
a
b
,
{\displaystyle \|\mathbf {a\times b} \|=ab\sin \theta _{ab}\ ,}
and the dot product:
a
⋅
b
=
a
b
cos
θ
a
b
,
{\displaystyle \mathbf {a\cdot b} =ab\cos \theta _{ab}\ ,}
where a = b = 1 for the unit sphere, results in the identity among the angles attributed to Gauss:
sin
θ
a
b
sin
θ
c
d
cos
x
=
cos
θ
a
c
cos
θ
b
d
−
cos
θ
a
d
cos
θ
b
c
,
{\displaystyle \sin \theta _{ab}\sin \theta _{cd}\cos x=\cos \theta _{ac}\cos \theta _{bd}-\cos \theta _{ad}\cos \theta _{bc}\ ,}
where x is the angle between a × b and c × d , or equivalently, between the planes defined by these vectors.[2]
^ a b Lyle Frederick Albright (2008). "§2.5.1 Vector algebra" . Albright's chemical engineering handbook . CRC Press. p. 68. ISBN 978-0-8247-5362-7 ^ a b c Gibbs & Wilson 1901 , pp. 77 ff ^
Francis Begnaud Hildebrand (1992). Methods of applied mathematics ISBN 0-486-67002-3
^ a b Richard Courant, Fritz John (2000). "Areas of parallelograms and volumes of parallelepipeds in higher dimensions" . Introduction to calculus and analysis, Volume II (Reprint of original 1974 Interscience ed.). Springer. pp. 190–195. ISBN 3-540-66569-2 ^ Gibbs & Wilson 1901 , §42 of section "Direct and skew products of vectors", p.77^ a b Gibbs & Wilson 1901 , p. 76^ Gibbs & Wilson 1901 , p. 77^ Gibbs & Wilson 1901 , Equation 27, p. 77^ Vidwan Singh Soni (2009). "§1.10.2 Vector quadruple product" . Mechanics and relativity . PHI Learning Pvt. Ltd. pp. 11–12. ISBN 978-81-203-3713-8 ^ This formula is applied to spherical trigonometry by Edwin Bidwell Wilson, Josiah Willard Gibbs (1901). "§42 in Direct and skew products of vectors ". Vector analysis: a text-book for the use of students of mathematics 77 ff .
^ "linear algebra - Cross-product identity" . Mathematics Stack Exchange . Retrieved 2021-10-07 .^ Joseph George Coffin (1911). Vector analysis: an introduction to vector-methods and their various applications to physics and mathematics 56 .
![{\displaystyle \det[\mathbf {A} ,\mathbf {B} ,\mathbf {C} ]=|\mathbf {A} ,\mathbf {B} ,\mathbf {C} |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbfcf1712714fb56603c5bc2458d5c0daf27a24)
![{\displaystyle (\mathbf {a\times b} )\mathbf {\times } (\mathbf {c} \times \mathbf {d} )=[\mathbf {a,\ b,\ d} ]\mathbf {c} -[\mathbf {a,\ b,\ c} ]\mathbf {d} \ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83e41e3db9c5e3a2fb2030f9822ab0af1f24ad91)
![{\displaystyle [\mathbf {a,\ b,\ c} ]=\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aef3f5f3c5d0bb976d5ae3b8e1ccd43861fdbc11)
![{\displaystyle [\mathbf {b,\ c,\ d} ]\mathbf {a} -[\mathbf {c,\ d,\ a} ]\mathbf {b} +[\mathbf {d,\ a,\ b} ]\mathbf {c} -[\mathbf {a,\ b,\ c} ]\mathbf {d} =0\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aab120a97d92e8ffd4c390fc8be37ebdf564021)
