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:<math> (\mathbf{a \times b} )\mathbf{\times} (\mathbf{c}\times \mathbf{d}) = \mathbf c - \mathbf d \ ,</math>
:<math> (\mathbf{a \times b} )\mathbf{\times} (\mathbf{c}\times \mathbf{d}) = \mathbf c - \mathbf d \ ,</math>
using the notation for the ]:
using the notation for the ]:
:<math> = (\mathbf{a \times b}) \mathbf{\cdot d } = \begin{vmatrix} \mathbf{a\cdot }\hat {\mathbf i} & \mathbf{a \cdot} \hat {\mathbf j} & \mathbf{a\cdot} \hat {\mathbf k}\\
:<math> = (\mathbf{a \times b}) \mathbf{\cdot d } = \begin{vmatrix} \mathbf{a\cdot }\hat {\mathbf i} & \mathbf{b \cdot} \hat {\mathbf i } & \mathbf{d \cdot} \hat {\mathbf i }\\
\mathbf{b\cdot }\hat {\mathbf i} & \mathbf{b\cdot} \hat {\mathbf j} & \mathbf{b\cdot}\hat {\mathbf k}\\ \mathbf{d\cdot} \hat {\mathbf i} & \mathbf{d\cdot} \hat {\mathbf j} & \mathbf{d\cdot }\hat {\mathbf k} \end{vmatrix}\ ,</math>
\mathbf{a \cdot }\hat {\mathbf j } & \mathbf{b\cdot} \hat {\mathbf j} & \mathbf{d \cdot}\hat {\mathbf j }\\ \mathbf{a \cdot} \hat {\mathbf k } & \mathbf{b \cdot} \hat {\mathbf k } & \mathbf{d\cdot }\hat {\mathbf k} \end{vmatrix} = \begin{vmatrix} \mathbf{a\cdot }\hat {\mathbf i} & \mathbf{a \cdot} \hat {\mathbf j} & \mathbf{a\cdot} \hat {\mathbf k}\\
where the last form is a determinant with <math> \hat {\mathbf i}, \ \hat {\mathbf j}, \ \hat {\mathbf k} </math> denoting unit vectors along three mutually orthogonal directions. \mathbf{b\cdot }\hat {\mathbf i} & \mathbf{b\cdot} \hat {\mathbf j} & \mathbf{b\cdot} \hat {\mathbf k }\\ \mathbf{d\cdot} \hat {\mathbf i } & \mathbf{d\cdot} \hat {\mathbf j } & \mathbf{d\cdot }\hat {\mathbf k} \end{vmatrix}
\ ,</math>
where the last two forms are determinants with <math> \hat {\mathbf i}, \ \hat {\mathbf j}, \ \hat {\mathbf k} </math> denoting unit vectors along three mutually orthogonal directions.
Equivalent forms can be obtained using the identity:
Equivalent forms can be obtained using the identity:
Revision as of 17:34, 16 September 2010
See also: Vector algebra relations
In mathematics , the quadruple product vectors in three-dimensional Euclidean space . The name "quadruple product" is used for two different products, the scalar-valued scalar quadruple product and the vector-valued vector quadruple product .
Scalar quadruple product
The scalar quadruple product is defined as the dot product of two cross products :
(
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⋅
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,
{\displaystyle (\mathbf {a\times b} )\mathbf {\cdot } (\mathbf {c} \times \mathbf {d} )\ ,}
where a, b, c, d are vectors in three-dimensional Euclidean space. It can be evaluated using the identity:
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⋅
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=
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{\displaystyle (\mathbf {a\times b} )\mathbf {\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 :
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{\displaystyle (\mathbf {a\times b} )\mathbf {\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
The vector quadruple product is defined as the cross product of two cross products:
(
a
×
b
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×
(
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. It can be evaluated using the identity:
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a
×
b
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×
(
c
×
d
)
=
[
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]
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{\displaystyle (\mathbf {a\times b} )\mathbf {\times } (\mathbf {c} \times \mathbf {d} )=\mathbf {c} -\mathbf {d} \ ,}
using the notation for the triple product :
[
a
,
b
,
d
]
=
(
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b
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⋅
d
=
|
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⋅
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=
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,
{\displaystyle =(\mathbf {a\times b} )\mathbf {\cdot d} ={\begin{vmatrix}\mathbf {a\cdot } {\hat {\mathbf {i} }}&\mathbf {b\cdot } {\hat {\mathbf {i} }}&\mathbf {d\cdot } {\hat {\mathbf {i} }}\\\mathbf {a\cdot } {\hat {\mathbf {j} }}&\mathbf {b\cdot } {\hat {\mathbf {j} }}&\mathbf {d\cdot } {\hat {\mathbf {j} }}\\\mathbf {a\cdot } {\hat {\mathbf {k} }}&\mathbf {b\cdot } {\hat {\mathbf {k} }}&\mathbf {d\cdot } {\hat {\mathbf {k} }}\end{vmatrix}}={\begin{vmatrix}\mathbf {a\cdot } {\hat {\mathbf {i} }}&\mathbf {a\cdot } {\hat {\mathbf {j} }}&\mathbf {a\cdot } {\hat {\mathbf {k} }}\\\mathbf {b\cdot } {\hat {\mathbf {i} }}&\mathbf {b\cdot } {\hat {\mathbf {j} }}&\mathbf {b\cdot } {\hat {\mathbf {k} }}\\\mathbf {d\cdot } {\hat {\mathbf {i} }}&\mathbf {d\cdot } {\hat {\mathbf {j} }}&\mathbf {d\cdot } {\hat {\mathbf {k} }}\end{vmatrix}}\ ,}
where the last two forms are determinants with
i
^
,
j
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,
k
^
{\displaystyle {\hat {\mathbf {i} }},\ {\hat {\mathbf {j} }},\ {\hat {\mathbf {k} }}}
Equivalent forms can be obtained using the identity:
[
b
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,
d
]
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−
[
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,
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]
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−
[
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0
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{\displaystyle \mathbf {a} -\mathbf {b} -\mathbf {d} =0\ .}
Interpretation
The quadruple products are useful for deriving various formulas in spherical and plane geometry.
References
^ Edwin Bidwell Wilson, Josiah Willard Gibbs (1901). "§42: Direct and skew products of vectors". Vector analysis: a text-book for the use of students of mathematics ff .
See also
Categories :
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denoting unit vectors along three mutually orthogonal directions.
