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| In [[mathematics]], the '''dot product''' is an operation that takes two [[vector]]s as input, and that returns a [[scalar]] [[number]] as output. The number returned is dependent on the length of both vectors, and on the angle between them. The name is derived from the [[Interpunct|centered dot]] "·" that is often used to designate this operation; the alternative name '''scalar product''' emphasizes the [[scalar (mathematics)|scalar]] (rather than [[Euclidean vector|vector]]) nature of the result.
| | *每道题目的解答都要有<font color="red" size=4>完整的解题过程</font>。中英文不限。 |
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| The dot product contrasts (in three dimensional space) with the [[cross product]], which produces a vector as result.
| | == Problem 1 == |
| | #有<math>k</math>种不同的明信片,每种明信片有无限多张,寄给<math>n</math>个人,每人一张,有多少种方法? |
| | #有<math>k</math>种不同的明信片,每种明信片有无限多张,寄给<math>n</math>个人,每人一张,每个人必须收到不同种类的明信片,有多少种方法? |
| | #有<math>k</math>种不同的明信片,每种明信片有无限多张,寄给<math>n</math>个人,每人收到<math>r</math>张不同的明信片(但不同的人可以收到相同的明信片),有多少种方法? |
| | #只有一种明信片,共有<math>m</math>张,寄给<math>n</math>个人,全部寄完,每个人可以收多张明信片或者不收明信片,有多少种方法? |
| | #有<math>k</math>种不同的明信片,其中第<math>i</math>种明信片有<math>m_i</math>张,寄给<math>n</math>个人,全部寄完,每个人可以收多张明信片或者不收明信片,有多少种方法? |
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| ==Definition== | | == Problem 2 == |
| The dot product of two vectors '''a''' = [''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub>] and '''b''' = [''b''<sub>1</sub>, ''b''<sub>2</sub>, ..., ''b''<sub>''n''</sub>] is defined as:
| | Find the number of ways to select <math>2n</math> balls from <math>n</math> identical blue balls, <math>n</math> identical red balls and <math>n</math> identical green balls. |
| :<math>\mathbf{a}\cdot \mathbf{b} = \sum_{i=1}^n a_ib_i = a_1b_1 + a_2b_2 + \cdots + a_nb_n </math>
| | * Give a combinatorial proof for the problem. |
| where Σ denotes [[Summation|summation notation]] ( the sum of all the terms) and ''n'' is the dimension of the vector space.
| | * Give an algebraic proof for the problem. |
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| In dimension 2, the dot product of vectors [a,b] and [c,d] is ac + bd.
| | == Problem 3 == |
| The same way, in a dimension 3, the dot product of vectors [a,b,c] and [d,e,f] is ad + be + cf.
| | *一个长度为<math>n</math>的“山峦”是如下由<math>n</math>个"/"和<math>n</math>个"\"组成的,从坐标<math>(0,0)</math>到<math>(0,2n)</math>的折线,但任何时候都不允许低于<math>x</math>轴。例如下图: |
| For example, the dot product of two three-dimensional vectors [1, 3, −5] and [4, −2, −1] is
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| :<math>
| | /\ |
| [1, 3, -5] \cdot [4, -2, -1] = (1 \times 4) + (3 \times (-2)) + ((-5) \times (-1)) = (4) - (6) + (5) = 3.
| | / \/\/\ /\/\ |
| </math> | | / \/\/ \/\/\ |
| | ---------------------- |
| | :长度为<math>n</math>的“山峦”有多少? |
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| ==Geometric interpretation==
| | *一个长度为<math>n</math>的“地貌”是由<math>n</math>个"/"和<math>n</math>个"\"组成的,从坐标<math>(0,0)</math>到<math>(0,2n)</math>的折线,允许低于<math>x</math>轴。长度为<math>n</math>的“地貌”有多少? |
| [[File:Dot Product.svg|thumb|300px|right|'''A''' • '''B''' = <nowiki>|</nowiki>'''A'''<nowiki>|</nowiki> <nowiki>|</nowiki>'''B'''<nowiki>|</nowiki> cos(θ). <br /><nowiki>|</nowiki>'''A'''<nowiki>|</nowiki> cos(θ) is the [[scalar resolute|scalar projection]] of '''A''' onto '''B'''.]]
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| In [[Euclidean geometry]], the dot product, [[Euclidean norm|length]], and [[angle]] are related. For a vector '''a''', the dot product '''a''' · '''a''' is the square of the length of '''a''', or
| | == Problem 4== |
| | 李雷和韩梅梅竞选学生会主席,韩梅梅获得选票 <math>p</math> 张,李雷获得选票 <math>q</math> 张,<math>p>q</math>。我们将总共的 <math>p+q</math> 张选票一张一张的点数,有多少种选票的排序方式使得在整个点票过程中,韩梅梅的票数一直高于李雷的票数?等价地,假设选票均匀分布的随机排列,以多大概率在整个点票过程中,韩梅梅的票数一直高于李雷的票数。 |
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| :<math>{\mathbf{a} \cdot \mathbf{a}}=\left\|\mathbf{a}\right\|^2</math>
| | ==Problem 5== |
| | A <math>2\times n</math> rectangle is to be paved with <math>1\times 2</math> identical blocks and <math>2\times 2</math> identical blocks. Let <math>f(n)</math> denote the number of ways that can be done. Find a recurrence relation for <math>f(n)</math>, solve the recurrence relation. |
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| where ||'''a'''|| denotes the [[Euclidean norm|length]] (magnitude) of '''a'''. More generally, if '''b''' is another vector
| | == Problem 6 == |
| | * 令<math>s_n</math>表示长度为<math>n</math>,没有2个连续的1的二进制串的数量,即 |
| | *:<math>s_n=|\{x\in\{0,1\}^n\mid \forall 1\le i\le n-1, x_ix_{i+1}\neq 11\}|</math>。 |
| | :求 <math>s_n</math>。 |
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| :<math> \mathbf{a} \cdot \mathbf{b}=\left\|\mathbf{a}\right\| \, \left\|\mathbf{b}\right\| \cos \theta \,</math> | | *令<math>t_n</math>表示长度为<math>n</math>,没有3个连续的1的二进制串的数量,即 |
| | *:<math>t_n=|\{x\in\{0,1\}^n\mid \forall 1\le i\le n-2, x_ix_{i+1}x_{i+2}\neq 111\}|</math>。 |
| | *#给出计算<math>t_n</math>的递归式,并给出足够的初始值。 |
| | *#计算<math>t_n</math>的生成函数<math>T(x)=\sum_{n\ge 0}t_n x^n</math>,给出生成函数<math>T(x)</math>的闭合形式。 |
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| where ||'''a'''|| and ||'''b'''|| denote the length of '''a''' and '''b''' and ''θ'' is the [[angle]] between them.
| | 注意:只需解生成函数的闭合形式,无需展开。 |
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| This formula can be rearranged to determine the size of the angle between two nonzero vectors:
| | == Problem 7 == |
| | | Let <math>a_n</math> be a sequence of numbers satisfying the recurrence relation: |
| :<math>\theta=\arccos \left( \frac {\bold{a}\cdot\bold{b}} {\left\|\bold{a}\right\|\left\|\bold{b}\right\|}\right)</math>
| | :<math>p a_n+q a_{n-1}+r a_{n-2}=0</math> |
| | | with initial condition <math>a_0=s</math> and <math>a_1=t</math>, where <math>p,q,r,s,t</math> are constants such that <math>{p}+q+r=0</math>, <math>p\neq 0</math> and <math>s\neq t</math>. Solve the recurrence relation. |
| One can also first convert the vectors to [[unit vector]]s by dividing by their magnitude:
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| :<math>\boldsymbol{\hat{a}} = \frac{\bold{a}}{\left\|\bold{a}\right\|}</math>
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| then the angle ''θ'' is given by
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| :<math>\theta = \arccos ( \boldsymbol{\hat a}\cdot\boldsymbol{\hat b})</math>
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| | |
| As the [[cosine]] of 90° is zero, the dot product of two [[orthogonal]](perpendicular) vectors is always zero. Moreover, two vectors can be considered [[orthogonal]] if and only if their dot product is zero, and they both have a nonzero length. This property provides a simple method to test the condition of orthogonality.
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| Sometimes these properties are also used for ''defining'' the dot product, especially in 2 and 3 dimensions; this definition is equivalent to the above one. For higher dimensions the formula can be used to define the concept of angle.
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| The geometric properties rely on the [[basis (linear algebra)|basis]] being [[orthonormal]], i.e. composed of pairwise perpendicular vectors with unit length.
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| ===Scalar projection===
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| If both '''a''' and '''b''' have length one (i.e., they are [[unit vector]]s), their dot product simply gives the cosine of the angle between them.
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| If only '''b''' is a [[unit vector]], then the dot product '''a''' '''·''' '''b''' gives |'''a'''| cos(θ), i.e., the magnitude of the projection of '''a''' in the direction of '''b''', with a minus sign if the direction is opposite. This is called the [[scalar resolute|scalar projection]] of '''a''' onto '''b''', or [[vector (geometry)|scalar component]] of '''a''' in the direction of '''b''' (see figure). This property of the dot product has several useful applications (for instance, see next section).
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| If neither '''a''' nor '''b''' is a unit vector, then the magnitude of the projection of '''a''' in the direction of '''b''', for example, would be '''a''' '''·''' ('''b''' / |'''b'''|) as the unit vector in the direction of '''b''' is '''b''' / |'''b'''|.
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| | |
| ===Rotation===
| |
| A [[rotation (mathematics)|rotation]] of the orthonormal basis in terms of which vector '''a''' is represented is obtained with a multiplication of '''a''' by a [[rotation matrix]] '''R'''. This [[matrix multiplication]] is just a compact representation of a sequence of dot products.
| |
| | |
| For instance, let
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| *''B''<sub>1</sub> = {'''x''', '''y''', '''z'''} and ''B''<sub>2</sub> = {'''u''', '''v''', '''w'''} be two different [[orthonormal basis|orthonormal bases]] of the same space '''R'''<sup>3</sup>, with ''B''<sub>2</sub> obtained by just rotating ''B''<sub>1</sub>,
| |
| *'''a'''<sub>1</sub> = (a<sub>x</sub>, a<sub>y</sub>, a<sub>z</sub>) represent vector '''a''' in terms of ''B''<sub>1</sub>,
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| * '''a'''<sub>2</sub> = (a<sub>u</sub>, a<sub>v</sub>, a<sub>w</sub>) represent the same vector in terms of the rotated basis ''B''<sub>2</sub>,
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| *'''u'''<sub>1</sub>, '''v'''<sub>1</sub>, '''w'''<sub>1</sub> be the rotated basis vectors '''u''', '''v''', '''w''' represented in terms of ''B''<sub>1</sub>.
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| Then the rotation from ''B''<sub>1</sub> to ''B''<sub>2</sub> is performed as follows:
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| | |
| :<math> \bold a_2 = \bold{Ra}_1 =
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| \begin{bmatrix} u_x & u_y & u_z \\ v_x & v_y & v_z \\ w_x & w_y & w_z \end{bmatrix}
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| \begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix} =
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| \begin{bmatrix} \bold u_1\cdot\bold a_1 \\ \bold v_1\cdot\bold a_1 \\ \bold w_1\cdot\bold a_1 \end{bmatrix} = \begin{bmatrix} a_u \\ a_v \\ a_w \end{bmatrix} .
| |
| </math> | |
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| Notice that the rotation matrix '''R''' is assembled by using the rotated basis vectors '''u'''<sub>1</sub>, '''v'''<sub>1</sub>, '''w'''<sub>1</sub> as its rows, and these vectors are unit vectors. By definition, '''Ra'''<sub>1</sub> consists of a sequence of dot products between each of the three rows of '''R''' and vector '''a'''<sub>1</sub>. Each of these dot products determines a scalar component of '''a''' in the direction of a rotated basis vector (see previous section).
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| If '''a'''<sub>1</sub> is a [[row vector]], rather than a [[column vector]], then '''R''' must contain the rotated basis vectors in its columns, and must post-multiply '''a'''<sub>1</sub>:
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| :<math> \bold a_2 = \bold a_1 \bold R = | |
| \begin{bmatrix} a_x & a_y & a_z \end{bmatrix}
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| \begin{bmatrix} u_x & v_x & w_x \\ u_y & v_y & w_y \\ u_z & v_z & w_z \end{bmatrix} =
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| \begin{bmatrix} \bold u_1\cdot\bold a_1 & \bold v_1\cdot\bold a_1 & \bold w_1\cdot\bold a_1 \end{bmatrix} = \begin{bmatrix} a_u & a_v & a_w \end{bmatrix} .
| |
| </math>
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| ==Physics==
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| In [[physics]], magnitude is a [[scalar (physics)|scalar]] in the physical sense, i.e. a [[physical quantity]] independent of the coordinate system, expressed as the [[product (mathematics)|product]] of a [[number|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.
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| Example:
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| * [[Mechanical work]] is the dot product of [[Force (physics)|force]] and [[Displacement (vector)|displacement]] vectors.
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| * [[Magnetic flux]] is the dot product of the [[magnetic field]] and the [[Area vector|area]] vectors.
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| ==Properties==
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| The following properties hold if '''a''', '''b''', and '''c''' are real [[vector (geometry)|vectors]] and ''r'' is a [[scalar (mathematics)|scalar]].
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| The dot product is [[commutative]]:
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| :<math> \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}.</math>
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| The dot product is [[distributive]] over vector addition:
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| :<math> \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}. </math>
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| The dot product is [[bilinear form|bilinear]]:
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| :<math> \mathbf{a} \cdot (r\mathbf{b} + \mathbf{c})
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| = r(\mathbf{a} \cdot \mathbf{b}) +(\mathbf{a} \cdot \mathbf{c}).
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| </math>
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| When multiplied by a scalar value, dot product satisfies:
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| :<math> (c_1\mathbf{a}) \cdot (c_2\mathbf{b}) = (c_1c_2) (\mathbf{a} \cdot \mathbf{b}) </math>
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| (these last two properties follow from the first two).
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| Two non-zero vectors '''a''' and '''b''' are [[perpendicular]] [[if and only if]] '''a''' • '''b''' = 0.
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| 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]]:
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| : 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'''.
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| Provided that the basis is orthonormal, the dot product is invariant under isometric changes of the basis: rotations, reflections, and combinations, keeping the origin fixed. The above mentioned geometric interpretation relies on this property. In other words, for an orthonormal space with any number of dimensions, the dot product is invariant under a [[coordinate transformation]] based on an [[orthogonal matrix]]. This corresponds to the following two conditions:
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| *The new basis is again orthonormal (i.e., it is orthonormal expressed in the old one).
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| *The new base vectors have the same length as the old ones (i.e., unit length in terms of the old basis).
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| If '''a''' and '''b''' are functions, then the derivative of '''a''' • '''b''' is '''a'''' • '''b''' + '''a''' • '''b''''
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| ==Triple product expansion==
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| {{Main|Triple product}}
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| This is a very useful identity (also known as '''Lagrange's formula''') involving the dot- and [[Cross product|cross-products]]. It is written as
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| :<math>\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a}\cdot\mathbf{c}) - \mathbf{c}(\mathbf{a}\cdot\mathbf{b})</math>
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| which is [[mnemonic|easier to remember]] as "BAC minus CAB", keeping in mind which vectors are dotted together. This formula is commonly used to simplify vector calculations in [[physics]].
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| ==Proof of the geometric interpretation==
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| Consider the element of '''R'''<sup>n</sup>
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| :<math> \mathbf{v} = v_1 \mathbf{\hat{e}}_1 + v_2 \mathbf{\hat{e}}_2 + ... + v_n \mathbf{\hat{e}}_n. \, </math>
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| Repeated application of the [[Pythagorean theorem]] yields for its length |'''v'''|
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| :<math> |\mathbf{v}|^2 = v_1^2 + v_2^2 + ... + v_n^2. \,</math>
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| But this is the same as
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| :<math> \mathbf{v} \cdot \mathbf{v} = v_1^2 + v_2^2 + ... + v_n^2, \,</math>
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| so we conclude that taking the dot product of a vector '''v''' with itself yields the squared length of the vector.
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| ; '''[[Lemma (mathematics)|Lemma]] 1''':<math> \mathbf{v} \cdot \mathbf{v} = |\mathbf{v}|^2. \, </math>
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| Now consider two vectors '''a''' and '''b''' extending from the origin, separated by an angle θ. A third vector '''c''' may be defined as
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| :<math> \mathbf{c} \ \stackrel{\mathrm{def}}{=}\ \mathbf{a} - \mathbf{b}. \,</math>
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| creating a triangle with sides '''a''', '''b''', and '''c'''. According to the [[law of cosines]], we have
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| :<math> |\mathbf{c}|^2 = |\mathbf{a}|^2 + |\mathbf{b}|^2 - 2 |\mathbf{a}||\mathbf{b}| \cos \theta. \,</math>
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| Substituting dot products for the squared lengths according to Lemma 1, we get
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| :<math>
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| \mathbf{c} \cdot \mathbf{c}
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| = \mathbf{a} \cdot \mathbf{a}
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| + \mathbf{b} \cdot \mathbf{b} | |
| - 2 |\mathbf{a}||\mathbf{b}| \cos\theta. \, | |
| </math> ''(1)''
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| But as '''c''' ≡ '''a''' − '''b''', we also have
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| :<math>
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| \mathbf{c} \cdot \mathbf{c}
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| = (\mathbf{a} - \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) \,</math>, | |
| which, according to the [[distributive law]], expands to
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| :<math>
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| \mathbf{c} \cdot \mathbf{c}
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| = \mathbf{a} \cdot \mathbf{a} | |
| + \mathbf{b} \cdot \mathbf{b}
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| -2(\mathbf{a} \cdot \mathbf{b}). \,
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| </math> ''(2)'' | |
| Merging the two '''c''' • '''c''' equations, ''(1)'' and ''(2)'', we obtain
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| :<math>
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| \mathbf{a} \cdot \mathbf{a}
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| + \mathbf{b} \cdot \mathbf{b}
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| -2(\mathbf{a} \cdot \mathbf{b})
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| = \mathbf{a} \cdot \mathbf{a} | |
| + \mathbf{b} \cdot \mathbf{b}
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| - 2 |\mathbf{a}||\mathbf{b}| \cos\theta. \,
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| </math> | |
| Subtracting '''a''' • '''a''' + '''b''' • '''b''' from both sides and dividing by −2 leaves
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| :<math> \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}| \cos\theta. \, </math>
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| [[Q.E.D.]]
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| ==Generalization==
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| <!-- Generalization can also include the idea of a scalar product using tensor notation. See the talk page under "more general definition". -->
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| The [[Inner product space|inner product]] generalizes the dot product to [[vector space|abstract vector spaces]] and is usually denoted by <math>\langle\mathbf{a}\, , \mathbf{b}\rangle</math>. Due to the geometric interpretation of the dot product the [[norm (mathematics)|norm]] ||'''a'''|| of a vector '''a''' in such an [[inner product space]] is defined as
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| :<math>\|\mathbf{a}\| = \sqrt{\langle\mathbf{a}\, , \mathbf{a}\rangle}</math>
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| such that it generalizes length, and the angle θ between two vectors '''a''' and '''b''' by | |
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| :<math> \cos{\theta} = \frac{\langle\mathbf{a}\, , \mathbf{b}\rangle}{\|\mathbf{a}\| \, \|\mathbf{b}\|}. </math>
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| In particular, two vectors are considered [[orthogonal]] if their inner product is zero
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| :<math> \langle\mathbf{a}\, , \mathbf{b}\rangle = 0.</math>
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| For vectors with complex entries, using the given definition of the dot product would lead to quite different geometric properties. For instance the dot product of a vector with itself can be an arbitrary complex number, and can be zero without the vector being the zero vector; this in turn would have severe consequences for notions like length and angle. Many geometric properties can be salvaged, at the cost of giving up the symmetric and bilinear properties of the scalar product, by alternatively defining
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| :<math>\mathbf{a}\cdot \mathbf{b} = \sum{a_i \overline{b_i}} </math>
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| where <span style="text-decoration: overline">''b<sub>i</sub>''</span> is the [[complex conjugate]] of ''b<sub>i</sub>''. 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 not linear in '''b''' (but rather [[conjugate linear]]), and the scalar product is not symmetric either, since
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| :<math> \mathbf{a} \cdot \mathbf{b} = \overline{\mathbf{b} \cdot \mathbf{a}} </math>.
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| This type of scalar product is nevertheless quite useful, and leads to the notions of [[Hermitian form]] and of general [[inner product space]]s.
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| The [[Frobenius inner product]] generalizes the dot product to matrices. It is defined as the sum of the products of the corresponding components of two matrices having the same size.
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| ===Generalization to tensors===
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| The dot product between a [[tensor]] of order n and a tensor of order m is a tensor of order n+m-2. The dot product is worked out by multiplying and summing across a single index in both tensors. If <math>\mathbf{A}</math> and <math>\mathbf{B}</math> are two tensors with element representation <math>A_{ij\dots}^{k\ell\dots}</math> and <math>B_{mn\dots}^{p{\dots}i}</math> the elements of the dot product <math>\mathbf{A} \cdot \mathbf{B}</math> are given by
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| :<math>A_{ij\dots}^{k\ell\dots}B_{mn\dots}^{p{\dots}i} = \sum_{i=1}^n A_{ij\dots}^{k\ell\dots}B_{mn\dots}^{p{\dots}i}</math>
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| This definition naturally reduces to the standard vector dot product when applied to vectors, and matrix multiplication when applied to matrices.
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| Occasionally, a double dot product is used to represent multiplying and summing across two indices. The double dot product between two 2nd order tensors is a scalar.
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| ==Related pages==
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| * [[Cauchy–Schwarz inequality]]
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| * [[Cross product]]
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| * [[Matrix multiplication]]
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| * [[Physics]]
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| ==Other websites==
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| * {{mathworld|urlname=DotProduct|title=Dot product}}
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| * [http://behindtheguesses.blogspot.com/2009/04/dot-and-cross-products.html A quick geometrical derivation and interpretation of dot product]
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| * [http://xahlee.org/SpecialPlaneCurves_dir/ggb/Vector_Dot_Product.html Interactive GeoGebra Applet]
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| * [http://www.falstad.com/dotproduct/ Java demonstration of dot product]
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| * [http://www.cs.brown.edu/exploratories/freeSoftware/repository/edu/brown/cs/exploratories/applets/dotProduct/dot_product_guide.html Another Java demonstration of dot product]
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| * [http://www.mathreference.com/la,dot.html Explanation of dot product including with complex vectors]
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| * [http://demonstrations.wolfram.com/DotProduct/ "Dot Product"] by Bruce Torrence, [[Wolfram Demonstrations Project]], 2007.
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| * Intuitive explanation [https://www.youtube.com/watch?v=_WgRwRyssk0 video 1] and [https://www.youtube.com/watch?v=YyNnK0T0w9o video 2] from online [https://www.udacity.com/course/interactive-3d-graphics--cs291 Interactive 3D graphics course]
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| [[Category:Linear algebra]]
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