???? ??? ????? ??????? ????, ???? ??????? ??? ????? ??????? (???? ?????? ?? ?????????? ?????? ?? ??? ??? ?? ???? ?????? ??? ????????? ???? ????? ??), ????-?????????? ???? (R3) ???? ?? ??????? ???? ??? ?????? (??-?????) ??????? ????? ?? ??? ????? ?????? × ??? ????? ????? ??? ?? ????? ??? '?? ???????? ????? a ??? b ????? ??? ?? ???? ??????? a × b, ??? ????? ????? ????? ?? ????? ??? ???????????? (?????) ????? ?? ??? ?? ?? ?? ????? ??? ???? ???? ???? (????) ??? ????? (90 ????? ??) ????? ??? ?? ???? ???? , ????? ?????? , ??????????? , ??? ??????? ??????????? ???? ???? ?????? ??????????? (?????) ??? ????? ??? ??????? ( ?????????? ?????? ) ???? ????? ???????

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???? ??????? ??? ?? ??????? ????? ????????? ???? ????? ?? ^{[1] [2]}

${\displaystyle \mathbf {a} \times \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\sin \theta \ \mathbf {n} }$

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???? ??????? ??? ???? ??? '?? ?? ?????? ?? ???? ?? ^{[note 1]} determinant :

${\displaystyle \mathbf {u\times v} ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\u_{1}&u_{2}&u_{3}\\v_{1}&v_{2}&v_{3}\\\end{vmatrix}}}$

?? ??????????? ??? ?????? ?? ???? ?? ???? ???? ??? ??? ???? ?? ???? ?? ??? ??????? ?????? ?? ???? ??? ?? ??? ???? ?? ???? ?? ?????? ???? ???? ????? ???, ?? ?? ?????? ????? ??

${\displaystyle {\begin{aligned}\mathbf {u\times v} &=\\(u_{2}v_{3}\mathbf {i} +u_{3}v_{1}\mathbf {j} +u_{1}v_{2}\mathbf {k} )-(u_{3}v_{2}\mathbf {i} +u_{1}v_{3}\mathbf {j} +u_{2}v_{1}\mathbf {k} )\\&=\\(u_{2}v_{3}-u_{3}v_{2})\mathbf {i} +(u_{3}v_{1}-u_{1}v_{3})\mathbf {j} +(u_{1}v_{2}-u_{2}v_{1})\mathbf {k} .\end{aligned}}}$

??????? ????? ????? ??? ????? ???? ?? ??? ?? ????? ?? ???, ?? ?? ?????? ????? ?? ^[3]

${\displaystyle \mathbf {u\times v} ={\begin{vmatrix}u_{2}&u_{3}\\v_{2}&v_{3}\end{vmatrix}}\mathbf {i} -{\begin{vmatrix}u_{1}&u_{3}\\v_{1}&v_{3}\end{vmatrix}}\mathbf {j} +{\begin{vmatrix}u_{1}&u_{2}\\v_{1}&v_{2}\end{vmatrix}}\mathbf {k} }$

?? ????? ??????? ?? ????? ????? ?? ?? ????? ??

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???????????? ??????? ?? ?????? ???? ???? ?? ??-????? ??????? ?? ???? ?? ????? ??, ??? ???? ???? ??????? ?? ?? ???? ????? ??:

${\displaystyle {\frac {d}{dt}}(\mathbf {a} \times \mathbf {b} )={\frac {d\mathbf {a} }{dt}}\times \mathbf {b} +\mathbf {a} \times {\frac {d\mathbf {b} }{dt}}}$

????? a ??? b ?? ????? ?? ?? ??????? ???????? t ???? ????? ???? ???

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???? [ ???? ]

• Cajori, Florian (1929). A History Of Mathematical Notations Volume II . Open Court Publishing . p. 134. ISBN   978-0-486-67766-8 . {{ cite book }} : Invalid |ref=harv ( help )
• E. A. Milne (1948) Vectorial Mechanics , Chapter 2: Vector Product, pp 11 –31, London: Methuen Publishing .
• Wilson, Edwin Bidwell (1901). Vector Analysis: A text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs . Yale University Press . {{ cite book }} : Invalid |ref=harv ( help )
• T. Levi-Civita; U. Amaldi (1949). Lezioni di meccanica razionale (in Italian). Zanichelli editore. {{ cite book }} : Unknown parameter |city= ignored ( |location= suggested) ( help ) CS1 maint: unrecognized language ( link )

????? ???? [ ???? ]

• Hazewinkel, Michiel, ed. (2001), "Cross product" , ???? ?? ????????? , ???????, ISBN   978-1-55608-010-4
• Weisstein, Eric W. , ???????.html "???? ???????" , ????-????? .
• A quick geometrical derivation and interpretation of cross products
• C.A. Gonano and R.E. Zich (2014). Cross product in N Dimensions - the doublewedge product , Polytechnic University of Milan, Italy.
• Z.K. Silagadze (2002). Multi-dimensional vector product. Journal of Physics. A35, 4949 Archived 2015-09-05 at the Wayback Machine . (it is only possible in 7-D space)
• Real and Complex Products of Complex Numbers
• An interactive tutorial Archived 2006-04-24 at the Wayback Machine . created at Syracuse University ? (requires java )
• W. Kahan (2007). Cross-Products and Rotations in Euclidean 2- and 3-Space. University of California, Berkeley (PDF).