सदिश बीजगणित के सूत्र

नीचे दी गयीं सर्वसमिकाएँ त्रिविमीय युक्लीडीय अवकाश के सदिशों से सम्बंधित हैं^[1] इनमें से कुछ (किन्तु सभी नहीं) अधिक वीमा वाले सदिशों के लिये भी सत्य हैं। उदाहरण के लिये, दो सदिशों का 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।

${\displaystyle \|\mathbf {A} \|^{2}=A_{1}^{2}+A_{2}^{2}+A_{3}^{2}\ }$

यह परिमाण अदिश गुणनफल का प्रयोग करते हुए निम्नलिखित प्रकार से भी अभिव्यक्त किया जा सकता है-

${\displaystyle \|\mathbf {A} \|^{2}=(\mathbf {A\cdot A} )\ }$

${\displaystyle {\frac {\mathbf {A\cdot B} }{\|\mathbf {A} \|\|\mathbf {B} \|}}\leq 1\ }$; Cauchy–Schwarz inequality in three dimensions

${\displaystyle \|\mathbf {A+B} \|\leq \|\mathbf {A} \|+\|\mathbf {B} \|}$; the triangle inequality in three dimensions

${\displaystyle \|\mathbf {A-B} \|\geq \|\mathbf {A} \|-\|\mathbf {B} \|}$; the 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]

${\displaystyle \sin \theta ={\frac {\|\mathbf {A\times B} \|}{\|\mathbf {A} \|\|\mathbf {B} \|}}\ \ (-\pi <\theta \leq \pi )}$

To satisfy the right-hand rule, for positive θ, vector B is counter-clockwise from A, and for negative θ it is clockwise.

${\displaystyle \cos \theta ={\frac {\mathbf {A\cdot B} }{\|\mathbf {A} \|\|\mathbf {B} \|}}\ \ (-\pi <\theta \leq \pi )}$

Here the notation A × B denotes the vector cross product of vectors A and B. The Pythagorean trigonometric identity then provides:

${\displaystyle \|\mathbf {A\times B} \|^{2}+(\mathbf {A\cdot B} )^{2}=\|\mathbf {A} \|^{2}\|\mathbf {B} \|^{2}}$

If a vector A = (A_x, A_y, A_z) makes angles α, β, γ with an orthogonal set of x-, y- and z-axes, then:

${\displaystyle \cos \alpha ={\frac {A_{x}}{\sqrt {A_{x}^{2}+A_{y}^{2}+A_{z}^{2}}}}={\frac {A_{x}}{\|\mathbf {A} \|}}\ ,}$

and analogously for angles β, γ. Consequently:

${\displaystyle \mathbf {A} =\|\mathbf {A} \|\left(\cos \alpha \ {\hat {\mathbf {i} }}+\cos \beta \ {\hat {\mathbf {j} }}+\cos \gamma \ {\hat {\mathbf {k} }}\right)\ ,}$

with ${\displaystyle {\hat {\mathbf {i} }},\ {\hat {\mathbf {j} }},\ {\hat {\mathbf {k} }}}$ unit vectors along the axis directions.

The area Σ of a parallelogram with sides A and B containing the angle θ is:

${\displaystyle \Sigma =AB\ \sin \theta \ ,}$

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:

${\displaystyle \Sigma =\|\mathbf {A\times B} \|={\sqrt {\|\mathbf {A} \|^{2}\|\mathbf {B} \|^{2}-(\mathbf {A\cdot B} )^{2}}}\ .}$

The square of this expression is:^[3]

${\displaystyle \Sigma ^{2}=(\mathbf {A\cdot A} )(\mathbf {B\cdot B} )-(\mathbf {A\cdot B} )(\mathbf {B\cdot A} )=\Gamma (\mathbf {A} ,\ \mathbf {B} )\ ,}$

where Γ(A, B) is the Gram determinant of A and B defined by:

${\displaystyle \Gamma (\mathbf {A} ,\ \mathbf {B} )={\begin{vmatrix}\mathbf {A\cdot A} &\mathbf {A\cdot B} \\\mathbf {B\cdot A} &\mathbf {B\cdot B} \end{vmatrix}}\ .}$

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]

${\displaystyle V^{2}=\Gamma (\mathbf {A} ,\ \mathbf {B} ,\ \mathbf {C} )={\begin{vmatrix}\mathbf {A\cdot A} &\mathbf {A\cdot B} &\mathbf {A\cdot C} \\\mathbf {B\cdot A} &\mathbf {B\cdot B} &\mathbf {B\cdot C} \\\mathbf {C\cdot A} &\mathbf {C\cdot B} &\mathbf {C\cdot C} \end{vmatrix}}\ .}$

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]

• ${\displaystyle c(\mathbf {A} +\mathbf {B} )=c\mathbf {A} +c\mathbf {B} }$; distributivity of multiplication by a scalar and addition
• ${\displaystyle \mathbf {A} +\mathbf {B} =\mathbf {B} +\mathbf {A} }$; commutativity of addition
• ${\displaystyle \mathbf {A} +(\mathbf {B} +\mathbf {C} )=(\mathbf {A} +\mathbf {B} )+\mathbf {C} }$; associativity of addition
• ${\displaystyle \mathbf {A} \cdot \mathbf {B} =\mathbf {B} \cdot \mathbf {A} }$; commutativity of scalar (dot) product
• ${\displaystyle \mathbf {A} \times \mathbf {B} =\mathbf {-B} \times \mathbf {A} }$; anticommutativity of vector cross product
• ${\displaystyle \left(\mathbf {A} +\mathbf {B} \right)\cdot \mathbf {C} =\mathbf {A} \cdot \mathbf {C} +\mathbf {B} \cdot \mathbf {C} }$; distributivity of addition wrt scalar product
• ${\displaystyle \left(\mathbf {A} +\mathbf {B} \right)\times \mathbf {C} =\mathbf {A} \times \mathbf {C} +\mathbf {B} \times \mathbf {C} }$; distributivity of addition wrt vector cross product
• ${\displaystyle \mathbf {A} \cdot \left(\mathbf {B} \times \mathbf {C} \right)=\mathbf {B} \cdot \left(\mathbf {C} \times \mathbf {A} \right)=\left(\mathbf {A} \times \mathbf {B} \right)\cdot \mathbf {C} }$

${\displaystyle =\left|{\begin{array}{ccc}A_{x}&B_{x}&C_{x}\\A_{y}&B_{y}&C_{y}\\A_{z}&B_{z}&C_{z}\end{array}}\right|=[\mathbf {A,\ B,\ C} ]}$ ; scalar triple product

• ${\displaystyle \mathbf {A\times } \left(\mathbf {B} \times \mathbf {C} \right)=\left(\mathbf {A} \cdot \mathbf {C} \right)\mathbf {B} -\left(\mathbf {A} \cdot \mathbf {B} \right)\mathbf {C} }$; vector triple product
• ${\displaystyle \mathbf {\left(A\times B\right)\cdot } \left(\mathbf {C} \times \mathbf {D} \right)=\left(\mathbf {A} \cdot \mathbf {C} \right)\left(\mathbf {B} \cdot \mathbf {D} \right)-\left(\mathbf {B} \cdot \mathbf {C} \right)\left(\mathbf {A} \cdot \mathbf {D} \right)}$; Binet–Cauchy identity in three dimensions

In particular, when A = C and B = D, the above reduces to:

${\displaystyle \mathbf {(A\times B)\cdot (A\times B)=|A\times B|^{2}=(A\cdot A)(B\cdot B)-(A\cdot B)^{2}} }$; Lagrange's identity in three dimensions

• ${\displaystyle [\mathbf {A} ,\mathbf {B} ,\mathbf {C} ]\mathbf {D} =\left(\mathbf {A} \cdot \mathbf {D} \right)\left(\mathbf {B} \times \mathbf {C} \right)+\left(\mathbf {B} \cdot \mathbf {D} \right)\left(\mathbf {C} \times \mathbf {A} \right)+\left(\mathbf {C} \cdot \mathbf {D} \right)\left(\mathbf {A} \times \mathbf {B} \right)}$
• A vector quadruple product, which is also a vector, can be defined, which satisfies the following identities:^[4][5]

${\displaystyle (\mathbf {A} \times \mathbf {B} )\times (\mathbf {C} \times \mathbf {D} )=[\mathbf {A} ,\mathbf {B} ,\mathbf {D} ]\mathbf {C} -[\mathbf {A} ,\mathbf {B} ,\mathbf {C} ]\mathbf {D} =[\mathbf {A} ,\mathbf {C} ,\mathbf {D} ]\mathbf {B} -[\mathbf {B} ,\mathbf {C} ,\mathbf {D} ]\mathbf {A} }$

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]

${\displaystyle \mathbf {D} ={\frac {\mathbf {D\cdot (B\times C)} }{[\mathbf {A,\ B,\ C} ]}}\ \mathbf {A} +{\frac {\mathbf {D\cdot (C\times A)} }{[\mathbf {A,\ B,\ C} ]}}\ \mathbf {B} +{\frac {\mathbf {D\cdot (A\times B)} }{[\mathbf {A,\ B,\ C} ]}}\ \mathbf {C} \ .}$

1. ↑ ^{अ आ इ} See, for example, Lyle Frederick Albright (2008). "§2.5.1 Vector algebra". Albright's chemical engineering handbook. CRC Press. पृ॰ 68. आई॰ऍस॰बी॰ऍन॰ 0-8247-5362-3.
2. ↑ Francis Begnaud Hildebrand (1992). Methods of applied mathematics (Reprint of Prentice-Hall 1965 2nd संस्करण). Courier Dover Publications. पृ॰ 24. आई॰ऍस॰बी॰ऍन॰ 0-486-67002-3. मूल से 17 अगस्त 2017 को पुरालेखित. अभिगमन तिथि 3 जनवरी 2017.
3. ↑ ^{अ आ} Richard 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 संस्करण). Springer. पपृ॰ 190–195. आई॰ऍस॰बी॰ऍन॰ 3-540-66569-2.
4. ↑ Vidwan Singh Soni (2009). "§1.10.2 Vector quadruple product". Mechanics and relativity. PHI Learning Pvt. Ltd. पपृ॰ 11–12. आई॰ऍस॰बी॰ऍन॰ 81-203-3713-1.
5. ↑ 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. पपृ॰ 77 ff.
6. ↑ Joseph George Coffin (1911). Vector analysis: an introduction to vector-methods and their various applications to physics and mathematics (2nd संस्करण). Wiley. पृ॰ 56.

• सदिश अवकाश (वेक्तर स्पेस)
• ज्यामितीय बीजगणित (Geometric algebra)
• सदिश कैलकुलस की सर्वसमिकाएँ