नीचे दी गयीं सर्वसमिकाएँ त्रिविमीय युक्लीडीय अवकाश के सदिशों से सम्बंधित हैं[1] सदिश गुणनफल सभी विमाओं के लिये उपलब्ध नहीं है।
सदिश A का परिमाण, इस सदिश के तीन परस्पर लम्बवत दिशाओं में कम्पोनेन्ट के वर्गों के योग के वर्गमूल के बराबर होता है।
‖
A
‖
2
=
A
1
2
+
A
2
2
+
A
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2
{\displaystyle \|\mathbf {A} \|^{2}=A_{1}^{2}+A_{2}^{2}+A_{3}^{2}\ }
यह परिमाण अदिश गुणनफल का प्रयोग करते हुए निम्नलिखित प्रकार से भी अभिव्यक्त किया जा सकता है-
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A
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2
=
(
A
⋅
A
)
{\displaystyle \|\mathbf {A} \|^{2}=(\mathbf {A\cdot A} )\ }
A
⋅
B
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A
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B
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≤
1
{\displaystyle {\frac {\mathbf {A\cdot B} }{\|\mathbf {A} \|\|\mathbf {B} \|}}\leq 1\ }
Cauchy–Schwarz inequality in three dimensions
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+
B
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≤
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{\displaystyle \|\mathbf {A+B} \|\leq \|\mathbf {A} \|+\|\mathbf {B} \|}
triangle inequality in three dimensions
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−
B
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≥
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A
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−
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B
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{\displaystyle \|\mathbf {A-B} \|\geq \|\mathbf {A} \|-\|\mathbf {B} \|}
reverse triangle inequality Here the notation (A · B ) denotes the dot product of vectors A and B .
The vector product and the scalar product of two vectors define the angle between them, say θ:[1] [2]
sin
θ
=
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A
×
B
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B
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(
−
π
<
θ
≤
π
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{\displaystyle \sin \theta ={\frac {\|\mathbf {A\times B} \|}{\|\mathbf {A} \|\|\mathbf {B} \|}}\ \ (-\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
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A
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B
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(
−
π
<
θ
≤
π
)
{\displaystyle \cos \theta ={\frac {\mathbf {A\cdot B} }{\|\mathbf {A} \|\|\mathbf {B} \|}}\ \ (-\pi <\theta \leq \pi )}
Here the notation A × B denotes the vector cross product of vectors A and B .
The Pythagorean trigonometric identity then provides:
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B
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2
+
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⋅
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=
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2
{\displaystyle \|\mathbf {A\times B} \|^{2}+(\mathbf {A\cdot B} )^{2}=\|\mathbf {A} \|^{2}\|\mathbf {B} \|^{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
=
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x
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A
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,
{\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
=
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A
‖
(
cos
α
i
^
+
cos
β
j
^
+
cos
γ
k
^
)
,
{\displaystyle \mathbf {A} =\|\mathbf {A} \|\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
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B
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=
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−
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{\displaystyle \Sigma =\|\mathbf {A\times B} \|={\sqrt {\|\mathbf {A} \|^{2}\|\mathbf {B} \|^{2}-(\mathbf {A\cdot B} )^{2}}}\ .}
The square of this expression is:[3]
Σ
2
=
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A
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(
B
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−
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⋅
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B
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Γ
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,
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,
{\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
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B
)
=
|
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⋅
A
A
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B
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{\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 and C is given by the Gram determinant of the three vectors:[3]
V
2
=
Γ
(
A
,
B
,
C
)
=
|
A
⋅
A
A
⋅
B
A
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B
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B
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C
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C
⋅
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C
⋅
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|
.
{\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}}\ .}
This process can be extended to n -dimensions.
Some of the following algebraic relations refer to the dot product and the cross product of vectors. These relations can be found in a variety of sources, for example, see Albright.[1]
c
(
A
+
B
)
=
c
A
+
c
B
{\displaystyle c(\mathbf {A} +\mathbf {B} )=c\mathbf {A} +c\mathbf {B} }
A
+
B
=
B
+
A
{\displaystyle \mathbf {A} +\mathbf {B} =\mathbf {B} +\mathbf {A} }
A
+
(
B
+
C
)
=
(
A
+
B
)
+
C
{\displaystyle \mathbf {A} +(\mathbf {B} +\mathbf {C} )=(\mathbf {A} +\mathbf {B} )+\mathbf {C} }
A
⋅
B
=
B
⋅
A
{\displaystyle \mathbf {A} \cdot \mathbf {B} =\mathbf {B} \cdot \mathbf {A} }
A
×
B
=
−
B
×
A
{\displaystyle \mathbf {A} \times \mathbf {B} =\mathbf {-B} \times \mathbf {A} }
(
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} }
(
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+
B
)
×
C
=
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×
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+
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×
C
{\displaystyle \left(\mathbf {A} +\mathbf {B} \right)\times \mathbf {C} =\mathbf {A} \times \mathbf {C} +\mathbf {B} \times \mathbf {C} }
A
⋅
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B
×
C
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=
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⋅
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C
×
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{\displaystyle \mathbf {A} \cdot \left(\mathbf {B} \times \mathbf {C} \right)=\mathbf {B} \cdot \left(\mathbf {C} \times \mathbf {A} \right)=\left(\mathbf {A} \times \mathbf {B} \right)\cdot \mathbf {C} }
=
|
A
x
B
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C
x
A
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C
y
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z
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=
[
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{\displaystyle =\left|{\begin{array}{ccc}A_{x}&B_{x}&C_{x}\\A_{y}&B_{y}&C_{y}\\A_{z}&B_{z}&C_{z}\end{array}}\right|=[\mathbf {A,\ B,\ C} ]}
scalar triple product
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{\displaystyle \mathbf {A\times } \left(\mathbf {B} \times \mathbf {C} \right)=\left(\mathbf {A} \cdot \mathbf {C} \right)\mathbf {B} -\left(\mathbf {A} \cdot \mathbf {B} \right)\mathbf {C} }
vector triple product
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⋅
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×
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)
=
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−
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(
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{\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)}
Binet–Cauchy identity in three dimensionsIn particular, when A = C and B = D , the above reduces to:
(
A
×
B
)
⋅
(
A
×
B
)
=
|
A
×
B
|
2
=
(
A
⋅
A
)
(
B
⋅
B
)
−
(
A
⋅
B
)
2
{\displaystyle \mathbf {(A\times B)\cdot (A\times B)=|A\times B|^{2}=(A\cdot A)(B\cdot B)-(A\cdot B)^{2}} }
Lagrange's identity in three dimensions
[
A
,
B
,
C
]
D
=
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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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)
{\displaystyle [\mathbf {A} ,\mathbf {B} ,\mathbf {C} ]\mathbf {D} =\left(\mathbf {A} \cdot \mathbf {D} \right)\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)}
A vector quadruple product, which is also a vector, can be defined, which satisfies the following identities:[4] [5]
(
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×
B
)
×
(
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×
D
)
=
[
A
,
B
,
D
]
C
−
[
A
,
B
,
C
]
D
=
[
A
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C
,
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]
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} }
where [A, B, C ] is the scalar triple product A · (B × C) or the determinant of the matrix {A, B, C } with the components of these vectors as columns . Given three arbitrary vectors not on the same line, A, B, C , any other vector D can be expressed in terms of these as:[6]
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 (B\times C)} }{[\mathbf {A,\ B,\ C} ]}}\ \mathbf {A} +{\frac {\mathbf {D\cdot (C\times A)} }{[\mathbf {A,\ B,\ C} ]}}\ \mathbf {B} +{\frac {\mathbf {D\cdot (A\times B)} }{[\mathbf {A,\ B,\ C} ]}}\ \mathbf {C} \ .}
↑ अ आ इ See, for example, Lyle Frederick Albright (2008). "§2.5.1 Vector algebra". Albright's chemical engineering handbook आई॰ऍस॰बी॰ऍन॰ 0-8247-5362-3
↑ Francis Begnaud Hildebrand (1992). Methods of applied mathematics आई॰ऍस॰बी॰ऍन॰ 0-486-67002-3 पुरालेखित . अभिगमन तिथि 3 जनवरी 2017 . ↑ अ आ
Richard Courant, Fritz John (2000). "Areas of parallelograms and volumes of parallelepipeds in higher dimensions". Introduction to calculus and analysis, Volume II आई॰ऍस॰बी॰ऍन॰ 3-540-66569-2
↑
Vidwan Singh Soni (2009). "§1.10.2 Vector quadruple product". Mechanics and relativity आई॰ऍस॰बी॰ऍन॰ 81-203-3713-1
↑ 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 ff .
↑
Joseph George Coffin (1911). Vector analysis: an introduction to vector-methods and their various applications to physics and mathematics
![{\displaystyle =\left|{\begin{array}{ccc}A_{x}&B_{x}&C_{x}\\A_{y}&B_{y}&C_{y}\\A_{z}&B_{z}&C_{z}\end{array}}\right|=[\mathbf {A,\ B,\ C} ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d824bc44b609a1a82e0452d64c23015eb6ce3f)
![{\displaystyle [\mathbf {A} ,\mathbf {B} ,\mathbf {C} ]\mathbf {D} =\left(\mathbf {A} \cdot \mathbf {D} \right)\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d82861665cb9658077022bb5a2d191e2ea2b86fc)
![{\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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/37ca2f3444bdc28fbd509c8d7d1d41158b5112e7)
![{\displaystyle \mathbf {D} ={\frac {\mathbf {D\cdot (B\times C)} }{[\mathbf {A,\ B,\ C} ]}}\ \mathbf {A} +{\frac {\mathbf {D\cdot (C\times A)} }{[\mathbf {A,\ B,\ C} ]}}\ \mathbf {B} +{\frac {\mathbf {D\cdot (A\times B)} }{[\mathbf {A,\ B,\ C} ]}}\ \mathbf {C} \ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb7485cda0dcacb26b618b2757b3209bc2061363)