Formulas about vectors in three-dimensional Euclidean space
The following are important identities in vector algebra. Identities that involve the magnitude of a vector , 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.[nb 1][1]
Most of these relations can be dated to founder of vector calculus Josiah Willard Gibbs, if not earlier.[2]
Magnitudes
The magnitude of a vector A can be expressed using the dot product:
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]
Since A, B, C are three-dimensional vectors, this is equal to the square of the scalar triple product below.
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] the scalar-valued scalar quadruple product and the vector-valued vector quadruple product or vector product of four vectors.
In 3 dimensions, a vector D can be expressed in terms of basis vectors {A,B,C} as:[12]
Applications
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:
in conjunction with the relation for the magnitude of the cross product:
and the dot product:
where a = b = 1 for the unit sphere, results in the identity among the angles attributed to Gauss:
where x is the angle between a × b and c × d, or equivalently, between the planes defined by these vectors.[2]
^There is also a seven-dimensional cross product of vectors that relates to multiplication in the octonions, but it does not satisfy these three-dimensional identities.
References
^ a bLyle Frederick Albright (2008). "§2.5.1 Vector algebra". Albright's chemical engineering handbook. CRC Press. p. 68. ISBN978-0-8247-5362-7.
^ a b cGibbs & Wilson 1901, pp. 77 ff
^
Francis Begnaud Hildebrand (1992). Methods of applied mathematics (Reprint of Prentice-Hall 1965 2nd ed.). Courier Dover Publications. p. 24. ISBN0-486-67002-3.
^ a bRichard 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. ISBN3-540-66569-2.
^Gibbs & Wilson 1901, §42 of section "Direct and skew products of vectors", p.77
^ a bGibbs & 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. ISBN978-81-203-3713-8.
^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. Scribner. pp. 77ff.
^Joseph George Coffin (1911). Vector analysis: an introduction to vector-methods and their various applications to physics and mathematics (2nd ed.). Wiley. p. 56.
Further reading
Gibbs, Josiah Willard; Wilson, Edwin Bidwell (1901). Vector analysis: a text-book for the use of students of mathematics. Scribner.