Formulas about vectors in three-dimensional Euclidean space
The following are important identities in vector algebra . Identities that involve the magnitude of a vector
‖
A
‖
{\displaystyle \|\mathbf {A} \|}
, or the dot product (scalar product) of two vectors A ·B , apply to vectors in any dimension. Identities that use the cross product (vector product) A ×B are defined only in three dimensions.[1] (There is a seven-dimensional cross product , but the identities do not hold in seven dimensions.)
Magnitudes [ ]
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}}
Inequalities [ ]
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 |}}
Angles [ ]
The vector product and the scalar product of two vectors define the angle between them, say θ :[1] [2]
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} }}}
unit vectors along the axis directions.
Areas and volumes [ ]
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:[3]
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:[3]
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} |}
below.
This process can be extended to n -dimensions.
Addition and multiplication of vectors [ ]
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 {-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
)
{\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} )}
=
|
A
B
C
|
=
|
A
x
B
x
C
x
A
y
B
y
C
y
A
z
B
z
C
z
|
{\displaystyle =|\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} }
.
Binet-Cauchy identity :
(
A
×
B
)
⋅
(
C
×
D
)
=
(
A
⋅
C
)
(
B
⋅
D
)
−
(
B
⋅
C
)
(
A
⋅
D
)
{\displaystyle \mathbf {\left(A\times B\right)\cdot } \left(\mathbf {C} \times \mathbf {D} \right)=\left(\mathbf {A} \cdot \mathbf {C} \right)\left(\mathbf {B} \cdot \mathbf {D} \right)-\left(\mathbf {B} \cdot \mathbf {C} \right)\left(\mathbf {A} \cdot \mathbf {D} \right)}
.
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}}
.
Vector quadruple product :[4] [5]
(
A
×
B
)
×
(
C
×
D
)
=
|
A
B
D
|
C
−
|
A
B
C
|
D
=
|
A
C
D
|
B
−
|
B
C
D
|
A
{\displaystyle (\mathbf {A} \times \mathbf {B} )\times (\mathbf {C} \times \mathbf {D} )\ =\ |\mathbf {A} \,\mathbf {B} \,\mathbf {D} |\,\mathbf {C} \,-\,|\mathbf {A} \,\mathbf {B} \,\mathbf {C} |\,\mathbf {D} \ =\ |\mathbf {A} \,\mathbf {C} \,\mathbf {D} |\,\mathbf {B} \,-\,|\mathbf {B} \,\mathbf {C} \,\mathbf {D} |\,\mathbf {A} }
.
A consequence of the previous equation:[6]
|
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:[7]
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} .}
See also [ ]
References [ ]