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| {{See also|Vector calculus identities}}
| | Editor: Rather than acquiring a whole block of knives, of which you may well only use one particular or two, my recommendation would be to invest your gift card cash in a really great chef's knife. As it is, the set comes with a flimsy plastic storage box that is cumbersome for long-term use. Best Affordable Kitchen Knife Set We'll almost certainly chuck the box and retailer the knives in a knife block. If you have any questions relating to in which and how to use [http://www.thebestkitchenknivesreviews.com/ Best Electric Knife Sharpener America's Test Kitchen], you can make contact with us at our own web-site. We've continued to use the Victorinox set for the previous seven months, about after or twice a month. We did end up ditching the flimsy plastic box the set arrived in and rather keep them in a wooden knife block, which helps shield the blades. This is identified as the chefs knife bolster.<br><br>While this knife didn't stand out at first, by the end of the meal it subtly emerged as one particular of two of the knives that reduce the greatest. It was the only stamped straight-edge knife we tested, but it performed just as nicely or much better than the forged knives. Then we utilized the knife to vigorously reduce against a ceramic plate with about 35 swipes. However, following applying our honing steel, the knife was just as sharp as it was originally. This was also one of the only complaints about the set on Amazon.<br><br>I constantly thought a knife was a knife as long as it got the job accomplished! The least expensive and most ordinary knife is fantastic sufficient for me. But I can see how other persons might will need a superior knife. My wife's fond of cooking and obtaining the proper knife for the suitable cutting session assists a lot. I had no clue among the differences in knife blades and which ones are far more durable or which ones are less complicated to sharpen...now I will know the next time I'm searching to obtain a new set of kitchen knives!<br><br>Maintaining this in mind when deciding to buy the Cutco set will make a massive distinction to your cooking expertise. I know that our [https://Www.Gov.uk/search?q=original+buy original buy] in the late 70's had met all the challenges of daily use and the steak knife we utilised for our guest place settings have been the ideal for cutting meats and other foods we serve to them. There are claims by other companies to be the greatest, but the Cutco knife proved to me that it was the very best.<br><br>A very lightweight but sturdy folding knife that border patrol agents and cops will in fact carry with them is, at the finish of the day, a very lightweight and sturdy folding knife that quite a few folks from all walks of life carry and use each and every day (far from a combat knife that only a choose couple of definitely use). The Cold Steel GI Tanto Knife is a tactical knife, pure and uncomplicated - designed to be a lethal tiny friend to soldiers and other military operatives. |
| The relations below apply to [[Euclidean vector|vectors]] in a three-dimensional [[Euclidean space]].<ref name=Albright>See, for example, {{cite book |title=Albright's chemical engineering handbook |author=Lyle Frederick Albright |url=http://books.google.com/books?id=HYB3Udjx_FYC&pg=PA68 |page=68 |isbn=0-8247-5362-3 |publisher=CRC Press |chapter=§2.5.1 Vector algebra |year=2008}}
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| </ref> Some, but not all of them, extend to vectors of higher dimensions. In particular, the cross product of two vectors is not available in all dimensions. See [[Seven-dimensional cross product]].
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| ==Magnitudes==
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| The magnitude of a vector '''A''' is determined by its three components along three orthogonal directions using [[Pythagoras' theorem]]:
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| :<math>\|\mathbf A \|^2 = A_1^2 + A_2^2 +A_3^2 \ </math>
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| The magnitude also can be expressed using the [[dot product]]:
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| :<math>\|\mathbf A \|^2 = (\mathbf {A \cdot A}) \ </math>
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| ==Inequalities==
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| :<math>\frac{ \mathbf{A \cdot B}}{\|\mathbf A \| \|\mathbf B \|} \le 1 \ </math>; [[Cauchy–Schwarz inequality]] in three dimensions
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| :<math>\|\mathbf{A + B}\| \le \| \mathbf{A}\| + \|\mathbf{B}\| </math>; the [[triangle inequality]] in three dimensions
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| :<math>\|\mathbf{A - B}\| \ge \| \mathbf{A}\| - \|\mathbf{B}\| </math>; the [[Triangle_inequality#Reverse_triangle_inequality |reverse triangle inequality]]
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| Here the notation ('''A · B''') denotes the [[dot product]] of vectors '''A''' and '''B'''.
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| ==Angles==
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| The vector product and the scalar product of two vectors define the angle between them, say θ:<ref name=Albright/><ref name=Hildebrand>
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| {{cite book |title=Methods of applied mathematics |author=Francis Begnaud Hildebrand |page=24 |url=http://books.google.com/?id=17EZkWPz_eQC&pg=PA24|isbn=0-486-67002-3 |edition=Reprint of Prentice-Hall 1965 2nd|publisher=Courier Dover Publications |year=1992}}
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| </ref>
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| :<math>\sin \theta =\frac{\|\mathbf{A \times B}\|}{\|\mathbf A \| \|\mathbf B \|} \ \ ( -\pi < \theta \le \pi ) </math>
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| To satisfy the [[right-hand rule]], for positive θ, vector '''B''' is counter-clockwise from '''A''', and for negative θ it is clockwise.
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| :<math>\cos \theta = \frac{ \mathbf{A \cdot B}}{\|\mathbf A \| \|\mathbf B \|} \ \ ( -\pi < \theta \le \pi )</math>
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| Here the notation '''A × B''' denotes the vector [[cross product]] of vectors '''A''' and '''B'''.
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| The [[Pythagorean trigonometric identity]] then provides:
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| :<math> \|\mathbf{A \times B}\|^2 +(\mathbf{A \cdot B})^2 = \|\mathbf A \|^2 \|\mathbf B \|^2 </math>
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| If a vector '''A''' = (''A<sub>x</sub>, A<sub>y</sub>, A<sub>z</sub>'') makes angles α, β, γ with an orthogonal set of ''x-'', ''y-'' and ''z-''axes, then:
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| :<math> \cos \alpha = \frac{ A_x }{ \sqrt {A_x^2 +A_y^2 +A_z^2} } = \frac {A_x} {\| \mathbf A \|} \ , </math>
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| and analogously for angles β, γ. Consequently:
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| :<math>\mathbf A = \|\mathbf A \|\left( \cos \alpha \ \hat{\mathbf i} + \cos \beta\ \hat{\mathbf j} + \cos \gamma \ \hat{\mathbf k} \right) \ ,</math>
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| with <math>\hat{\mathbf i}, \ \hat{\mathbf j}, \ \hat{\mathbf k}</math> unit vectors along the axis directions.
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| ==Areas and volumes==
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| The area Σ of a [[parallelogram]] with sides ''A'' and ''B'' containing the angle θ is:
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| :<math> \Sigma = AB \ \sin \theta \ , </math>
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| 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:
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| :<math>\Sigma = \|\mathbf { A \times B } \| = \sqrt{ \|\mathbf A\|^2 \|\mathbf B\|^2 -(\mathbf{A \cdot B} )^2} \ . </math>
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| The square of this expression is:<ref name=Courant>
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| {{cite book |title=Introduction to calculus and analysis, Volume II |author=Richard Courant, Fritz John |url=http://books.google.com/books?id=ngkQxS4eicgC&pg=PA191 |pages=190–195 |chapter=Areas of parallelograms and volumes of parallelpipeds in higher dimensions |isbn=3-540-66569-2 |year=2000 |publisher=Springer |edition=Reprint of original 1974 Interscience}}
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| </ref>
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| :<math>\Sigma^2 = (\mathbf{A \cdot A })(\mathbf{B \cdot B })-(\mathbf{A \cdot B })(\mathbf{B \cdot A })=\Gamma(\mathbf A,\ \mathbf B ) \ , </math>
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| where Γ('''A''', '''B''') is the [[Gram determinant]] of '''A''' and '''B''' defined by:
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| :<math>\Gamma(\mathbf A,\ \mathbf B )=\begin{vmatrix} \mathbf{A\cdot A} & \mathbf{A\cdot B} \\
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| \mathbf{B\cdot A} & \mathbf{B\cdot B} \end{vmatrix} \ . </math>
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| In a similar fashion, the squared volume ''V'' of a [[parallelpiped]] spanned by the three vectors '''A''', '''B''' and '''C''' is given by the Gram determinant of the three vectors:<ref name=Courant/>
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| :<math>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}\\
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| \mathbf{C\cdot A} & \mathbf{C\cdot B} & \mathbf{C\cdot C} \end{vmatrix} \ . </math>
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| This process can be extended to ''n''-dimensions.
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| ==Addition and multiplication of vectors==
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| 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.<ref name=Albright/>
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| *<math> c (\mathbf{A}+\mathbf{B})=c\mathbf{A}+c\mathbf{B} </math>; distributivity of multiplication by a scalar and addition
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| *<math> \mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A} </math>; commutativity of addition
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| *<math> \mathbf{A}+(\mathbf{B}+\mathbf{C})=(\mathbf{A}+\mathbf{B})+\mathbf{C} </math>; associativity of addition
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| *<math> \mathbf{A}\cdot\mathbf{B}=\mathbf{B}\cdot\mathbf{A} </math>; commutativity of scalar (dot) product
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| *<math> \mathbf{A}\times\mathbf{B}=\mathbf{-B}\times\mathbf{A} </math>; anticommutativity of vector cross product
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| *<math> \left(\mathbf{A}+\mathbf{B}\right)\cdot\mathbf{C}=\mathbf{A}\cdot\mathbf{C}+\mathbf{B}\cdot\mathbf{C} </math>; distributivity of addition wrt scalar product
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| *<math> \left(\mathbf{A}+\mathbf{B}\right)\times\mathbf{C}=\mathbf{A}\times\mathbf{C}+\mathbf{B}\times\mathbf{C} </math>; distributivity of addition wrt vector cross product
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| *<math> \mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{C}\right)=\mathbf{B}\cdot\left(\mathbf{C}\times\mathbf{A}\right)=\mathbf{C}\cdot\left(\mathbf{A}\times\mathbf{B}\right)</math>
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| ::::<math>=\left|\begin{array}{ccc}
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| A_{x} & B_{x} & C_{x}\\
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| A_{y} & B_{y} & C_{y}\\
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| A_{z} & B_{z} & C_{z}\end{array}\right| = [\mathbf{A, \ B,\ C }] </math> ; [[scalar triple product]]
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| *<math> \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} </math>; [[vector triple product]]
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| *<math> \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) </math>; [[Binet–Cauchy identity]] in three dimensions
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| :In particular, when '''A''' = '''C''' and '''B''' = '''D''', the above reduces to:
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| ::<math>\mathbf{(A \times B) \cdot (A \times B) = |A \times B|^2 = (A \cdot A) (B \cdot B)-(A \cdot B)^2 } </math>; [[Lagrange's identity]] in three dimensions
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| *<math>[\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)</math>
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| *A vector quadruple product, which is also a vector, can be defined, which satisfies the following identities:<ref name=Soni>
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| {{cite book |title=Mechanics and relativity |author=Vidwan Singh Soni |url=http://books.google.com/books?id=-3H5V0LGBOgC&pg=PA11 |pages=11–12 |chapter=§1.10.2 Vector quadruple product |publisher=PHI Learning Pvt. Ltd. |isbn=81-203-3713-1 |year=2009}}
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| </ref><ref name=Gibbs>This formula is applied to spherical trigonometry by | |
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| {{cite book |title=Vector analysis: a text-book for the use of students of mathematics |author=Edwin Bidwell Wilson, Josiah Willard Gibbs |url=http://books.google.com/books?id=RC8PAAAAIAAJ&pg=PA77 |chapter=§42 in ''Direct and skew products of vectors'' |publisher=Scribner |year=1901 |pages=77 ''ff''}}
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| </ref>
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| : <math>(\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}=
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| [\mathbf{A},\mathbf{C}, \mathbf{D}]\mathbf{B}-[\mathbf{B}, \mathbf{C},\mathbf{D}]\mathbf{A}</math>
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| :where ['''A, B, C'''] is the scalar [[triple product]] '''A · (B × C)''' or the [[determinant]] of the [[matrix (mathematics)|matrix]] {'''A, B, C'''} with the components of these vectors as columns .
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| *Given three arbitrary vectors not on the same line, '''A, B, C''', any other vector '''D''' can be expressed in terms of these as:<ref name=Coffin>
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| {{cite book |title=Vector analysis: an introduction to vector-methods and their various applications to physics and mathematics |author=Joseph George Coffin |url=http://books.google.com/books?id=9mgGAQAAIAAJ&pg=PA56 |page=56 |year=1911 |publisher=Wiley |edition=2nd}}
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| </ref> | |
| :<math>\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 \ .</math>
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| ==References==
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| <references/>
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| ==See also==
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| *[[Vector space]]
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| *[[Geometric algebra]]
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| [[Category:Vectors]]
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| [[Category:Mathematical identities]]
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| [[Category:Mathematics-related lists]]
| |
Editor: Rather than acquiring a whole block of knives, of which you may well only use one particular or two, my recommendation would be to invest your gift card cash in a really great chef's knife. As it is, the set comes with a flimsy plastic storage box that is cumbersome for long-term use. Best Affordable Kitchen Knife Set We'll almost certainly chuck the box and retailer the knives in a knife block. If you have any questions relating to in which and how to use Best Electric Knife Sharpener America's Test Kitchen, you can make contact with us at our own web-site. We've continued to use the Victorinox set for the previous seven months, about after or twice a month. We did end up ditching the flimsy plastic box the set arrived in and rather keep them in a wooden knife block, which helps shield the blades. This is identified as the chefs knife bolster.
While this knife didn't stand out at first, by the end of the meal it subtly emerged as one particular of two of the knives that reduce the greatest. It was the only stamped straight-edge knife we tested, but it performed just as nicely or much better than the forged knives. Then we utilized the knife to vigorously reduce against a ceramic plate with about 35 swipes. However, following applying our honing steel, the knife was just as sharp as it was originally. This was also one of the only complaints about the set on Amazon.
I constantly thought a knife was a knife as long as it got the job accomplished! The least expensive and most ordinary knife is fantastic sufficient for me. But I can see how other persons might will need a superior knife. My wife's fond of cooking and obtaining the proper knife for the suitable cutting session assists a lot. I had no clue among the differences in knife blades and which ones are far more durable or which ones are less complicated to sharpen...now I will know the next time I'm searching to obtain a new set of kitchen knives!
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