Case 6:12-cv-00799-JRG Document 143-4 Filed 04/11/14 Page 1 of 9 PageID #: 4269
`Case 6:12—cv—00799—JRG Document 143-4 Filed 04/11/14 Page 1 of 9 Page|D #: 4269
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`EXHIBIT D
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`EXHIBIT D
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`Case 6:12—cv—00799—JRG Document 143-4 Filed 04/11/14 Page 1 of 9 Page|D #: 4269
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`EXHIBIT D
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`EXHIBIT D
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`Dot product - Wikipedia, the free encyclopedia
`Page 1 of 8
`Case 6:12-cv-00799-JRG Document 143-4 Filed 04/11/14 Page 2 of 9 PageID #: 4270
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`Dot product
`
`From Wikipedia, the free encyclopedia
`
`In mathematics, the dot product, or scalar product (or sometimes inner product in the
`context of Euclidean space), is an algebraic operation that takes two equal-length sequences
`of numbers (usually coordinate vectors) and returns a single number. This operation can be
`defined either algebraically or geometrically. Algebraically, it is the sum of the products of the
`corresponding entries of the two sequences of numbers. Geometrically, it is the product of the
`magnitudes of the two vectors and the cosine of the angle between them. The name "dot
`product" is derived from the centered dot " · " that is often used to designate this operation; the
`alternative name "scalar product" emphasizes the scalar (rather than vectorial) nature of the
`result.
`
`In three-dimensional space, the dot product contrasts with the cross product of two vectors,
`which produces a pseudovector as the result. The dot product is directly related to the cosine
`of the angle between two vectors in Euclidean space of any number of dimensions.
`
`Contents
`
`■ 1 Definition
`■ 1.1 Algebraic definition
`■ 1.2 Geometric definition
`■ 1.3 Scalar projection and first properties
`■ 1.4 Equivalence of the definitions
`■ 2 Properties
`■ 2.1 Application to the cosine law
`■ 3 Triple product expansion
`■ 4 Physics
`■ 5 Generalizations
`■ 5.1 Complex vectors
`■ 5.2 Inner product
`■ 5.3 Functions
`■ 5.4 Weight function
`■ 5.5 Dyadics and matrices
`■ 5.6 Tensors
`■ 6 See also
`■ 7 References
`■ 8 External links
`
`Definition
`
`http://en.wikipedia.org/wiki/Dot_product
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`The dot product is often defined in one of two ways: algebraically or geometrically. The
`geometric definition is based on the notions of angle and distance (magnitude of vectors). The
`equivalence of these two definitions relies on having a Cartesian coordinate system for
`Euclidean space.
`
`In modern presentations of Euclidean geometry, the points of space are defined in terms of
`their Cartesian coordinates, and Euclidean space itself is commonly identified with the real
`coordinate space Rn. In such a presentation, the notions of length and angles are not primitive.
`They are defined by means of the dot product: the length of a vector is defined as the square
`root of the dot product of the vector by itself, and the cosine of the (non oriented) angle of two
`vectors of length one is defined as their dot product. So the equivalence of the two definitions
`of the dot product is a part of the equivalence of the classical and the modern formulations of
`Euclidean geometry.
`
`Algebraic definition
`
`The dot product of two vectors a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] is defined as:[1]
`
`where Σ denotes summation notation and n is the dimension of the vector space. For instance,
`in three-dimensional space, the dot product of vectors [1, 3, −5] and [4, −2, −1] is:
`
`Geometric definition
`
`In Euclidean space, a Euclidean vector is a geometrical object that possesses both a
`magnitude and a direction. A vector can be pictured as an arrow. Its magnitude is its length,
`and its direction is the direction the arrow points. The magnitude of a vector A is denoted by
`. The dot product of two Euclidean vectors A and B is defined by[2]
`
`where θ is the angle between A and B.
`
`In particular, if A and B are orthogonal, then the angle between them is 90° and
`
`At the other extreme, if they are codirectional, then the angle between them is 0° and
`
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`This implies that the dot product of a vector A by itself is
`
`which gives
`
`the formula for the Euclidean length of the vector.
`
`Scalar projection and first properties
`
`The scalar projection (or scalar component) of a Euclidean
`vector A in the direction of a Euclidean vector B is given by
`
`where θ is the angle between A and B.
`
`In terms of the geometric definition of the dot product, this
`can be rewritten
`
`where
`
`is the unit vector in the direction of B.
`
`The dot product is thus characterized geometrically by[3]
`
`Scalar projection
`
`The dot product, defined in this manner, is homogeneous
`under scaling in each variable, meaning that for any scalar
`α,
`
`It also satisfies a distributive law, meaning that
`
`Distributive law for the dot product
`
`These properties may be summarized by saying that the dot product is a bilinear form.
`Moreover, this bilinear form is positive definite, which means that
`is never negative and
`is zero if and only if
`
`Equivalence of the definitions
`
`If e1,...,en are the standard basis vectors in Rn, then we may write
`
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`The vectors ei are an orthonormal basis, which means that they have unit length and are at
`right angles to each other. Hence since these vectors have unit length
`
`and since they form right angles with each other, if i ≠ j,
`
`Now applying the distributivity of the geometric version of the dot product gives
`
`which is precisely the algebraic definition of the dot product. So the (geometric) dot product
`equals the (algebraic) dot product.
`
`Properties
`
`The dot product fulfils the following properties if a, b, and c are real vectors and r is a scalar.[1]
`[2]
`
`1. Commutative:
`
`which follows from the definition (θ is the angle between a and b):
`
`2. Distributive over vector addition:
`
`3. Bilinear:
`
`4. Scalar multiplication:
`
`5. Orthogonal:
`
`Two non-zero vectors a and b are orthogonal if and only if a ⋅ b = 0.
`
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`6. No cancellation:
`
`Unlike multiplication of ordinary numbers, where if ab = ac, then b always equals c
`unless a is zero, the dot product does not obey the cancellation law:
`If a ⋅ b = a ⋅ c and a ≠ 0, then we can write: a ⋅ (b − c) = 0 by the distributive law;
`the result above says this just means that a is perpendicular to (b − c), which still
`allows (b − c) ≠ 0, and therefore b ≠ c.
`
`7. Derivative: If a and b are functions, then the derivative (denoted by a prime ′) of a ⋅ b is
`a′ ⋅ b + a ⋅ b′.
`
`Application to the cosine law
`
`Main article: law of cosines
`
`Given two vectors a and b separated by angle θ (see image right), they form
`a triangle with a third side c = a − b. The dot product of this with itself is:
`
`Triangle with
`vector edges a
`
`and b,
`
`separated by
`
`angle θ.
`
`which is the law of cosines.
`
`Triple product expansion
`
`Main article: Triple product
`
`This is an identity (also known as Lagrange's formula) involving the dot- and cross-products.
`It is written as:[1][2]
`
`which is easier to remember as "BAC minus CAB", keeping in mind which vectors are dotted
`together. This formula finds application in simplifying vector calculations in physics.
`
`Physics
`
`In physics, vector magnitude is a scalar in the physical sense, i.e. a physical quantity
`independent of the coordinate system, expressed as the product of a numerical value and a
`physical unit, not just a number. The dot product is also a scalar in this sense, given by the
`formula, independent of the coordinate system. Examples include:[4][5]
`
`■ Mechanical work is the dot product of force and displacement vectors.
`
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`■ Magnetic flux is the dot product of the magnetic field and the area vectors.
`
`Generalizations
`
`Complex vectors
`
`For vectors with complex entries, using the given definition of the dot product would lead to
`quite different properties. For instance the dot product of a vector with itself would be an
`arbitrary complex number, and could be zero without the vector being the zero vector (such
`vectors are called isotropic); this in turn would have consequences for notions like length and
`angle. Properties such as the positive-definite norm can be salvaged at the cost of giving up
`the symmetric and bilinear properties of the scalar product, through the alternative definition[1]
`
`where bi is the complex conjugate of bi. Then the scalar product of any vector with itself is a
`non-negative real number, and it is nonzero except for the zero vector. However this scalar
`product is thus sesquilinear rather than bilinear: it is conjugate linear and not linear in b, and
`the scalar product is not symmetric, since
`
`The angle between two complex vectors is then given by
`
`This type of scalar product is nevertheless useful, and leads to the notions of Hermitian form
`and of general inner product spaces.
`
`Inner product
`
`Main article: Inner product space
`
`The inner product generalizes the dot product to abstract vector spaces over a field of scalars,
`being either the field of real numbers
`or the field of complex numbers
`. It is usually
`denoted by
`.
`
`The inner product of two vectors over the field of complex numbers is, in general, a complex
`number, and is sesquilinear instead of bilinear. An inner product space is a normed vector
`space, and the inner product of a vector with itself is real and positive-definite.
`
`Functions
`
`http://en.wikipedia.org/wiki/Dot_product
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`The dot product is defined for vectors that have a finite number of entries. Thus these vectors
`can be regarded as discrete functions: a length-n vector u is, then, a function with domain
`{k ∈ ℕ ∣ 1 ≤ k ≤ n}, and ui is a notation for the image of i by the function/vector u.
`
`This notion can be generalized to continuous functions: just as the inner product on vectors
`uses a sum over corresponding components, the inner product on functions is defined as an
`integral over some interval a ≤ x ≤ b (also denoted [a, b]):[1]
`
`Generalized further to complex functions ψ(x) and χ(x), by analogy with the complex inner
`product above, gives[1]
`
`Weight function
`
`Inner products can have a weight function, i.e. a function which weight each term of the inner
`product with a value.
`
`Dyadics and matrices
`
`Matrices have the Frobenius inner product, which is analogous to the vector inner product. It is
`defined as the sum of the products of the corresponding components of two matrices A and B
`having the same size:
`
`(For real matrices)
`
`Dyadics have a dot product and "double" dot product defined on them, see Dyadics (Product of
`dyadic and dyadic) for their definitions.
`
`Tensors
`
`The inner product between a tensor of order n and a tensor of order m is a tensor of order
`n + m − 2, see tensor contraction for details.
`
`See also
`
`■ Cauchy–Schwarz inequality
`■ Cross product
`
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`■ Matrix multiplication
`
`References
`
`a b c d e f
`
`S. Lipschutz, M. Lipson (2009). Linear Algebra (Schaum’s Outlines) (4th ed.). McGraw
`1. ^
`Hill. ISBN 978-0-07-154352-1.
`2. ^ a b c M.R. Spiegel, S. Lipschutz, D. Spellman (2009). Vector Analysis (Schaum’s Outlines) (2nd
`ed.). McGraw Hill. ISBN 978-0-07-161545-7.
`3. ^ Arfken, G. B.; Weber, H. J. (2000). Mathematical Methods for Physicists (5th ed.). Boston, MA:
`Academic Press. pp. 14–15. ISBN 978-0-12-059825-0..
`4. ^ K.F. Riley, M.P. Hobson, S.J. Bence (2010). Mathematical methods for physics and engineering
`(3rd ed.). Cambridge University Press. ISBN 978-0-521-86153-3.
`5. ^ M. Mansfield, C. O’Sullivan (2011). Understanding Physics (4th ed.). John Wiley & Sons.
`ISBN 978-0-47-0746370.
`
`External links
`
`■ Hazewinkel, Michiel, ed. (2001), "Inner
`product" (http://www.encyclopediaofmath.org/index.php?title=p/i051240), Encyclopedia
`of Mathematics, Springer, ISBN 978-1-55608-010-4
`■ Weisstein, Eric W., "Dot product (http://mathworld.wolfram.com/DotProduct.html)",
`MathWorld.
`■ Explanation of dot product including with complex vectors
`(http://www.mathreference.com/la,dot.html)
`■ "Dot Product" (http://demonstrations.wolfram.com/DotProduct/) by Bruce Torrence,
`Wolfram Demonstrations Project, 2007.
`
`Retrieved from "http://en.wikipedia.org/w/index.php?title=Dot_product&oldid=602821360"
`Categories: Bilinear forms Linear algebra Vectors Analytic geometry
`
`■ This page was last modified on 5 April 2014 at 05:07.
`■ Text is available under the Creative Commons Attribution-ShareAlike License; additional
`terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy.
`Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit
`organization.
`
`http://en.wikipedia.org/wiki/Dot_product
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`4/7/2014
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