Misplaced Pages

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90-462: When any two sine waves of the same frequency (but arbitrary phase ) are linearly combined , the result is another sine wave of the same frequency; this property is unique among periodic waves. Conversely, if some phase is chosen as a zero reference, a sine wave of arbitrary phase can be written as the linear combination of two sine waves with phases of zero and a quarter cycle, the sine and cosine components , respectively. A sine wave represents

180-493: A {\displaystyle {\color {red}\mathbf {a} }} and b {\displaystyle {\color {blue}\mathbf {b} }} separated by angle θ {\displaystyle \theta } (see the upper image ), they form a triangle with a third side c = a − b {\displaystyle {\color {orange}\mathbf {c} }={\color {red}\mathbf {a} }-{\color {blue}\mathbf {b} }} . Let

270-472: A {\displaystyle \mathbf {a} } and b {\displaystyle \mathbf {b} } is defined by a ⋅ b = ‖ a ‖ ‖ b ‖ cos ⁡ θ , {\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta ,} where θ {\displaystyle \theta }

360-413: A − b ) ⋅ ( a − b ) = a ⋅ a − a ⋅ b − b ⋅ a + b ⋅ b = a 2 − a ⋅ b −

450-1277: A ⋅ b + b 2 = a 2 − 2 a ⋅ b + b 2 c 2 = a 2 + b 2 − 2 a b cos ⁡ θ {\displaystyle {\begin{aligned}\mathbf {\color {orange}c} \cdot \mathbf {\color {orange}c} &=(\mathbf {\color {red}a} -\mathbf {\color {blue}b} )\cdot (\mathbf {\color {red}a} -\mathbf {\color {blue}b} )\\&=\mathbf {\color {red}a} \cdot \mathbf {\color {red}a} -\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} -\mathbf {\color {blue}b} \cdot \mathbf {\color {red}a} +\mathbf {\color {blue}b} \cdot \mathbf {\color {blue}b} \\&={\color {red}a}^{2}-\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} -\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} +{\color {blue}b}^{2}\\&={\color {red}a}^{2}-2\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} +{\color {blue}b}^{2}\\{\color {orange}c}^{2}&={\color {red}a}^{2}+{\color {blue}b}^{2}-2{\color {red}a}{\color {blue}b}\cos \mathbf {\color {purple}\theta } \\\end{aligned}}} which

540-542: A {\displaystyle a} , b {\displaystyle b} and c {\displaystyle c} denote the lengths of a {\displaystyle {\color {red}\mathbf {a} }} , b {\displaystyle {\color {blue}\mathbf {b} }} , and c {\displaystyle {\color {orange}\mathbf {c} }} , respectively. The dot product of this with itself is: c ⋅ c = (

630-445: A ‖ cos ⁡ θ i = a i , {\displaystyle \mathbf {a} \cdot \mathbf {e} _{i}=\left\|\mathbf {a} \right\|\,\left\|\mathbf {e} _{i}\right\|\cos \theta _{i}=\left\|\mathbf {a} \right\|\cos \theta _{i}=a_{i},} where a i {\displaystyle a_{i}} is the component of vector a {\displaystyle \mathbf {a} } in

720-404: A ⋅ e i ) = ∑ i b i a i = ∑ i a i b i , {\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} \cdot \sum _{i}b_{i}\mathbf {e} _{i}=\sum _{i}b_{i}(\mathbf {a} \cdot \mathbf {e} _{i})=\sum _{i}b_{i}a_{i}=\sum _{i}a_{i}b_{i},} which is precisely

810-536: A ⋅ b = 0. {\displaystyle \mathbf {a} \cdot \mathbf {b} =0.} At the other extreme, if they are codirectional , then the angle between them is zero with cos ⁡ 0 = 1 {\displaystyle \cos 0=1} and a ⋅ b = ‖ a ‖ ‖ b ‖ {\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\,\left\|\mathbf {b} \right\|} This implies that

900-527: A ⋅ c ) b − ( a ⋅ b ) c . {\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \cdot \mathbf {c} )\,\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\,\mathbf {c} .} This identity, also known as Lagrange's formula , may be remembered as "ACB minus ABC", keeping in mind which vectors are dotted together. This formula has applications in simplifying vector calculations in physics . In physics ,

990-490: A ⋅ c ) + α δ ( a ⋅ d ) + β γ ( b ⋅ c ) + β δ ( b ⋅ d ) . {\displaystyle =\alpha \gamma (\mathbf {a} \cdot \mathbf {c} )+\alpha \delta (\mathbf {a} \cdot \mathbf {d} )+\beta \gamma (\mathbf {b} \cdot \mathbf {c} )+\beta \delta (\mathbf {b} \cdot \mathbf {d} ).} Given two vectors

SECTION 10

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1080-407: A ⋅ ( γ c + δ d ) ) + β ( b ⋅ ( γ c + δ d ) ) = {\displaystyle =\alpha (\mathbf {a} \cdot (\gamma \mathbf {c} +\delta \mathbf {d} ))+\beta (\mathbf {b} \cdot (\gamma \mathbf {c} +\delta \mathbf {d} ))=} = α γ (

1170-413: A i b i ¯ , {\displaystyle \mathbf {a} \cdot \mathbf {b} =\sum _{i}{{a_{i}}\,{\overline {b_{i}}}},} where b i ¯ {\displaystyle {\overline {b_{i}}}} is the complex conjugate of b i {\displaystyle b_{i}} . When vectors are represented by column vectors ,

1260-430: A i b i = a 1 b 1 + a 2 b 2 + ⋯ + a n b n {\displaystyle \mathbf {a} \cdot \mathbf {b} =\sum _{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}} where Σ {\displaystyle \Sigma } denotes summation and n {\displaystyle n}

1350-434: A n ] {\displaystyle \mathbf {a} =[a_{1},a_{2},\cdots ,a_{n}]} and b = [ b 1 , b 2 , ⋯ , b n ] {\displaystyle \mathbf {b} =[b_{1},b_{2},\cdots ,b_{n}]} , specified with respect to an orthonormal basis , is defined as: a ⋅ b = ∑ i = 1 n

1440-471: A Euclidean vector is a geometric 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 to which the arrow points. The magnitude of a vector a {\displaystyle \mathbf {a} } is denoted by ‖ a ‖ {\displaystyle \left\|\mathbf {a} \right\|} . The dot product of two Euclidean vectors

1530-728: A compact subset K {\displaystyle K} of R n {\displaystyle \mathbb {R} ^{n}} with the standard Lebesgue measure , the above definition becomes: ⟨ f , g ⟩ = ∫ K f ( x ) g ( x ) d n ⁡ x . {\displaystyle \left\langle f,g\right\rangle =\int _{K}f(\mathbf {x} )g(\mathbf {x} )\,\operatorname {d} ^{n}\mathbf {x} .} Generalized further to complex continuous functions ψ {\displaystyle \psi } and χ {\displaystyle \chi } , by analogy with

1620-463: A field of scalars , being either the field of real numbers R {\displaystyle \mathbb {R} } or the field of complex numbers C {\displaystyle \mathbb {C} } . It is usually denoted using angular brackets by ⟨ a , b ⟩ {\displaystyle \left\langle \mathbf {a} \,,\mathbf {b} \right\rangle } . The inner product of two vectors over

1710-441: A specific range of frequencies . The audible frequency range for humans is typically given as being between about 20 Hz and 20,000 Hz (20 kHz), though the high frequency limit usually reduces with age. Other species have different hearing ranges. For example, some dog breeds can perceive vibrations up to 60,000 Hz. In many media, such as air, the speed of sound is approximately independent of frequency, so

1800-446: A zero at the origin of the complex frequency plane. The gain of its frequency response increases at a rate of +20  dB per decade of frequency (for root-power quantities), the same positive slope as a 1 order high-pass filter 's stopband , although a differentiator doesn't have a cutoff frequency or a flat passband . A n-order high-pass filter approximately applies the n time derivative of signals whose frequency band

1890-485: A flat passband. A n-order low-pass filter approximately performs the n time integral of signals whose frequency band is significantly higher than the filter's cutoff frequency. Frequency Frequency (symbol f ), most often measured in hertz (symbol: Hz), is the number of occurrences of a repeating event per unit of time . It is also occasionally referred to as temporal frequency for clarity and to distinguish it from spatial frequency . Ordinary frequency

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1980-411: A fractional error of Δ f f = 1 2 f T m {\textstyle {\frac {\Delta f}{f}}={\frac {1}{2fT_{\text{m}}}}} where T m {\displaystyle T_{\text{m}}} is the timing interval and f {\displaystyle f} is the measured frequency. This error decreases with frequency, so it

2070-473: A function with domain { k ∈ N : 1 ≤ k ≤ n } {\displaystyle \{k\in \mathbb {N} :1\leq k\leq n\}} , and u i {\displaystyle u_{i}} is a notation for the image of i {\displaystyle i} by the function/vector u {\displaystyle u} . This notion can be generalized to square-integrable functions : just as

2160-482: A heart beats at a frequency of 120 times per minute (2 hertz), the period—the time interval between beats—is half a second (60 seconds divided by 120). For cyclical phenomena such as oscillations , waves , or for examples of simple harmonic motion , the term frequency is defined as the number of cycles or repetitions per unit of time. The conventional symbol for frequency is f or ν (the Greek letter nu )

2250-416: A known frequency near the unknown frequency is mixed with the unknown frequency in a nonlinear mixing device such as a diode . This creates a heterodyne or "beat" signal at the difference between the two frequencies. If the two signals are close together in frequency the heterodyne is low enough to be measured by a frequency counter. This process only measures the difference between the unknown frequency and

2340-402: A repeating event is accomplished by counting the number of times that event occurs within a specific time period, then dividing the count by the period. For example, if 71 events occur within 15 seconds the frequency is: f = 71 15 s ≈ 4.73 Hz . {\displaystyle f={\frac {71}{15\,{\text{s}}}}\approx 4.73\,{\text{Hz}}.} If

2430-438: A single frequency with no harmonics and is considered an acoustically pure tone . Adding sine waves of different frequencies results in a different waveform. Presence of higher harmonics in addition to the fundamental causes variation in the timbre , which is the reason why the same musical pitch played on different instruments sounds different. Sine waves of arbitrary phase and amplitude are called sinusoids and have

2520-409: A single line. This could, for example, be considered the value of a wave along a wire. In two or three spatial dimensions, the same equation describes a travelling plane wave if position x {\displaystyle x} and wavenumber k {\displaystyle k} are interpreted as vectors, and their product as a dot product . For more complex waves such as the height of

2610-476: A water wave in a pond after a stone has been dropped in, more complex equations are needed. French mathematician Joseph Fourier discovered that sinusoidal waves can be summed as simple building blocks to approximate any periodic waveform, including square waves . These Fourier series are frequently used in signal processing and the statistical analysis of time series . The Fourier transform then extended Fourier series to handle general functions, and birthed

2700-491: Is homogeneous under scaling in each variable, meaning that for any scalar α {\displaystyle \alpha } , ( α a ) ⋅ b = α ( a ⋅ b ) = a ⋅ ( α b ) . {\displaystyle (\alpha \mathbf {a} )\cdot \mathbf {b} =\alpha (\mathbf {a} \cdot \mathbf {b} )=\mathbf {a} \cdot (\alpha \mathbf {b} ).} It also satisfies

2790-450: Is positive definite , which means that a ⋅ a {\displaystyle \mathbf {a} \cdot \mathbf {a} } is never negative, and is zero if and only if a = 0 {\displaystyle \mathbf {a} =\mathbf {0} } , the zero vector. If e 1 , ⋯ , e n {\displaystyle \mathbf {e} _{1},\cdots ,\mathbf {e} _{n}} are

Sine wave - Misplaced Pages Continue

2880-547: Is a non-negative real number, and it is nonzero except for the zero vector. However, the complex dot product is sesquilinear rather than bilinear, as it is conjugate linear and not linear in a {\displaystyle \mathbf {a} } . The dot product is not symmetric, since a ⋅ b = b ⋅ a ¯ . {\displaystyle \mathbf {a} \cdot \mathbf {b} ={\overline {\mathbf {b} \cdot \mathbf {a} }}.} The angle between two complex vectors

2970-435: Is also used. The period T is the time taken to complete one cycle of an oscillation or rotation. The frequency and the period are related by the equation f = 1 T . {\displaystyle f={\frac {1}{T}}.} The term temporal frequency is used to emphasise that the frequency is characterised by the number of occurrences of a repeating event per unit time. The SI unit of frequency

3060-535: Is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors ), and returns a single number. In Euclidean geometry , the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely the projection product ) of Euclidean space , even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more). Algebraically,

3150-495: Is called a radio wave . Likewise, an electromagnetic wave with a frequency higher than 8 × 10  Hz will also be invisible to the human eye; such waves are called ultraviolet (UV) radiation. Even higher-frequency waves are called X-rays , and higher still are gamma rays . All of these waves, from the lowest-frequency radio waves to the highest-frequency gamma rays, are fundamentally the same, and they are all called electromagnetic radiation . They all travel through vacuum at

3240-407: Is defined as the product of the projection of the first vector onto the second vector and the magnitude of the second vector. For example: 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 could be zero without the vector being the zero vector (e.g. this would happen with

3330-416: Is expressed with the unit reciprocal second (s ) or, in the case of radioactivity, with the unit becquerel . It is defined as a rate , f = N /Δ t , involving the number of entities counted or the number of events happened ( N ) during a given time duration (Δ t ); it is a physical quantity of type temporal rate . Dot product In mathematics , the dot product or scalar product

3420-411: Is generally a problem at low frequencies where the number of counts N is small. An old method of measuring the frequency of rotating or vibrating objects is to use a stroboscope . This is an intense repetitively flashing light ( strobe light ) whose frequency can be adjusted with a calibrated timing circuit. The strobe light is pointed at the rotating object and the frequency adjusted up and down. When

3510-402: Is red light, 800 THz ( 8 × 10  Hz ) is violet light, and between these (in the range 400–800 THz) are all the other colors of the visible spectrum . An electromagnetic wave with a frequency less than 4 × 10  Hz will be invisible to the human eye; such waves are called infrared (IR) radiation. At even lower frequency, the wave is called a microwave , and at still lower frequencies it

3600-476: Is related to angular frequency (symbol ω , with SI unit radian per second) by a factor of 2 π . The period (symbol T ) is the interval of time between events, so the period is the reciprocal of the frequency: T = 1/ f . Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals ( sound ), radio waves , and light . For example, if

3690-1181: Is significantly lower than the filter's cutoff frequency. Integrating any sinusoid with respect to time can be viewed as dividing its amplitude by its angular frequency and delaying it a quarter cycle: ∫ A sin ⁡ ( ω t + φ ) d t = − A ω cos ⁡ ( ω t + φ ) + C = − A ω sin ⁡ ( ω t + φ + π 2 ) + C = A ω sin ⁡ ( ω t + φ − π 2 ) + C . {\displaystyle {\begin{aligned}\int A\sin(\omega t+\varphi )dt&=-{\frac {A}{\omega }}\cos(\omega t+\varphi )+C\\&=-{\frac {A}{\omega }}\sin(\omega t+\varphi +{\tfrac {\pi }{2}})+C\\&={\frac {A}{\omega }}\sin(\omega t+\varphi -{\tfrac {\pi }{2}})+C\,.\end{aligned}}} The constant of integration C {\displaystyle C} will be zero if

Sine wave - Misplaced Pages Continue

3780-493: Is termed rotational frequency , is revolution per minute , abbreviated r/min or rpm. 60 rpm is equivalent to one hertz. As a matter of convenience, longer and slower waves, such as ocean surface waves , are more typically described by wave period rather than frequency. Short and fast waves, like audio and radio, are usually described by their frequency. Some commonly used conversions are listed below: For periodic waves in nondispersive media (that is, media in which

3870-509: Is the Kronecker delta . Also, by the geometric definition, for any vector e i {\displaystyle \mathbf {e} _{i}} and a vector a {\displaystyle \mathbf {a} } , we note that a ⋅ e i = ‖ a ‖ ‖ e i ‖ cos ⁡ θ i = ‖

3960-657: Is the angle between a {\displaystyle \mathbf {a} } and b {\displaystyle \mathbf {b} } . In particular, if the vectors a {\displaystyle \mathbf {a} } and b {\displaystyle \mathbf {b} } are orthogonal (i.e., their angle is π 2 {\displaystyle {\frac {\pi }{2}}} or 90 ∘ {\displaystyle 90^{\circ }} ), then cos ⁡ π 2 = 0 {\displaystyle \cos {\frac {\pi }{2}}=0} , which implies that

4050-408: Is the determinant of the matrix whose columns are the Cartesian coordinates of the three vectors. It is the signed volume of the parallelepiped defined by the three vectors, and is isomorphic to the three-dimensional special case of the exterior product of three vectors. The vector triple product is defined by a × ( b × c ) = (

4140-888: Is the dimension of the vector space . For instance, in three-dimensional space , the dot product of vectors [ 1 , 3 , − 5 ] {\displaystyle [1,3,-5]} and [ 4 , − 2 , − 1 ] {\displaystyle [4,-2,-1]} is:   [ 1 , 3 , − 5 ] ⋅ [ 4 , − 2 , − 1 ] = ( 1 × 4 ) + ( 3 × − 2 ) + ( − 5 × − 1 ) = 4 − 6 + 5 = 3 {\displaystyle {\begin{aligned}\ [1,3,-5]\cdot [4,-2,-1]&=(1\times 4)+(3\times -2)+(-5\times -1)\\&=4-6+5\\&=3\end{aligned}}} Likewise,

4230-641: Is the hertz (Hz), named after the German physicist Heinrich Hertz by the International Electrotechnical Commission in 1930. It was adopted by the CGPM (Conférence générale des poids et mesures) in 1960, officially replacing the previous name, cycle per second (cps). The SI unit for the period, as for all measurements of time, is the second . A traditional unit of frequency used with rotating mechanical devices, where it

4320-574: Is the law of cosines . There are two ternary operations involving dot product and cross product . The scalar triple product of three vectors is defined as a ⋅ ( b × c ) = b ⋅ ( c × a ) = c ⋅ ( a × b ) . {\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} ).} Its value

4410-400: Is the speed of light in vacuum, and this expression becomes f = c λ . {\displaystyle f={\frac {c}{\lambda }}.} When monochromatic waves travel from one medium to another, their frequency remains the same—only their wavelength and speed change. Measurement of frequency can be done in the following ways: Calculating the frequency of

4500-493: Is the unit vector in the direction of b {\displaystyle \mathbf {b} } . The dot product is thus characterized geometrically by a ⋅ b = a b ‖ b ‖ = b a ‖ a ‖ . {\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{b}\left\|\mathbf {b} \right\|=b_{a}\left\|\mathbf {a} \right\|.} The dot product, defined in this manner,

4590-602: Is the angle between a {\displaystyle \mathbf {a} } and b {\displaystyle \mathbf {b} } . In terms of the geometric definition of the dot product, this can be rewritten as a b = a ⋅ b ^ , {\displaystyle a_{b}=\mathbf {a} \cdot {\widehat {\mathbf {b} }},} where b ^ = b / ‖ b ‖ {\displaystyle {\widehat {\mathbf {b} }}=\mathbf {b} /\left\|\mathbf {b} \right\|}

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4680-418: Is then given by cos ⁡ θ = Re ⁡ ( a ⋅ b ) ‖ a ‖ ‖ b ‖ . {\displaystyle \cos \theta ={\frac {\operatorname {Re} (\mathbf {a} \cdot \mathbf {b} )}{\left\|\mathbf {a} \right\|\,\left\|\mathbf {b} \right\|}}.} The complex dot product leads to

4770-427: Is used for defining lengths (the length of a vector is the square root of the dot product of the vector by itself) and angles (the cosine of the angle between two vectors is the quotient of their dot product by the product of their lengths). The name "dot product" is derived from the dot operator "  ·  " that is often used to designate this operation; the alternative name "scalar product" emphasizes that

4860-401: The bounds of integration is an integer multiple of the sinusoid's period. An integrator has a pole at the origin of the complex frequency plane. The gain of its frequency response falls off at a rate of -20 dB per decade of frequency (for root-power quantities), the same negative slope as a 1 order low-pass filter 's stopband, although an integrator doesn't have a cutoff frequency or

4950-434: The distributive law , meaning that a ⋅ ( b + c ) = a ⋅ b + a ⋅ c . {\displaystyle \mathbf {a} \cdot (\mathbf {b} +\mathbf {c} )=\mathbf {a} \cdot \mathbf {b} +\mathbf {a} \cdot \mathbf {c} .} These properties may be summarized by saying that the dot product is a bilinear form . Moreover, this bilinear form

5040-476: The norm squared , a ⋅ a = ‖ a ‖ 2 {\textstyle \mathbf {a} \cdot \mathbf {a} =\|\mathbf {a} \|^{2}} , after the Euclidean norm ; it is a vector generalization of the absolute square of a complex scalar (see also: squared Euclidean distance ). The inner product generalizes the dot product to abstract vector spaces over

5130-1648: The standard basis vectors in R n {\displaystyle \mathbf {R} ^{n}} , then we may write a = [ a 1 , … , a n ] = ∑ i a i e i b = [ b 1 , … , b n ] = ∑ i b i e i . {\displaystyle {\begin{aligned}\mathbf {a} &=[a_{1},\dots ,a_{n}]=\sum _{i}a_{i}\mathbf {e} _{i}\\\mathbf {b} &=[b_{1},\dots ,b_{n}]=\sum _{i}b_{i}\mathbf {e} _{i}.\end{aligned}}} The vectors e i {\displaystyle \mathbf {e} _{i}} are an orthonormal basis , which means that they have unit length and are at right angles to each other. Since these vectors have unit length, e i ⋅ e i = 1 {\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{i}=1} and since they form right angles with each other, if i ≠ j {\displaystyle i\neq j} , e i ⋅ e j = 0. {\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{j}=0.} Thus in general, we can say that: e i ⋅ e j = δ i j , {\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{j}=\delta _{ij},} where δ i j {\displaystyle \delta _{ij}}

5220-495: The above example in this way, a 1 × 3 matrix ( row vector ) is multiplied by a 3 × 1 matrix ( column vector ) to get a 1 × 1 matrix that is identified with its unique entry: [ 1 3 − 5 ] [ 4 − 2 − 1 ] = 3 . {\displaystyle {\begin{bmatrix}1&3&-5\end{bmatrix}}{\begin{bmatrix}4\\-2\\-1\end{bmatrix}}=3\,.} In Euclidean space ,

5310-756: The algebraic definition of the dot product. So the geometric dot product equals the algebraic dot product. The dot product fulfills the following properties if a {\displaystyle \mathbf {a} } , b {\displaystyle \mathbf {b} } , c {\displaystyle \mathbf {c} } and d {\displaystyle \mathbf {d} } are real vectors and α {\displaystyle \alpha } , β {\displaystyle \beta } , γ {\displaystyle \gamma } and δ {\displaystyle \delta } are scalars . The commutative property can also be easily proven with

5400-601: The algebraic definition, and in more general spaces (where the notion of angle might not be geometrically intuitive but an analogous product can be defined) the angle between two vectors can be defined as θ = arccos ⁡ ( a ⋅ b ‖ a ‖ ‖ b ‖ ) . {\displaystyle \theta =\operatorname {arccos} \left({\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|}}\right).} = α (

5490-491: The alternating current in household electrical outlets is 60 Hz (between the tones B ♭ and B; that is, a minor third above the European frequency). The frequency of the ' hum ' in an audio recording can show in which of these general regions the recording was made. Aperiodic frequency is the rate of incidence or occurrence of non- cyclic phenomena, including random processes such as radioactive decay . It

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5580-422: The complex inner product above, gives: ⟨ ψ , χ ⟩ = ∫ K ψ ( z ) χ ( z ) ¯ d z . {\displaystyle \left\langle \psi ,\chi \right\rangle =\int _{K}\psi (z){\overline {\chi (z)}}\,{\text{d}}z.} Inner products can have a weight function (i.e.,

5670-423: The direction of e i {\displaystyle \mathbf {e} _{i}} . The last step in the equality can be seen from the figure. Now applying the distributivity of the geometric version of the dot product gives a ⋅ b = a ⋅ ∑ i b i e i = ∑ i b i (

5760-440: The dot product can also be written as a matrix product a ⋅ b = a T b , {\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{\mathsf {T}}\mathbf {b} ,} where a T {\displaystyle a{^{\mathsf {T}}}} denotes the transpose of a {\displaystyle \mathbf {a} } . Expressing

5850-447: The dot product can be expressed as a matrix product involving a conjugate transpose , denoted with the superscript H: a ⋅ b = b H a . {\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {b} ^{\mathsf {H}}\mathbf {a} .} In the case of vectors with real components, this definition is the same as in the real case. The dot product of any vector with itself

5940-414: The dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry , Euclidean spaces are often defined by using vector spaces . In this case, the dot product

6030-494: The dot product of a vector a {\displaystyle \mathbf {a} } with itself is a ⋅ a = ‖ a ‖ 2 , {\displaystyle \mathbf {a} \cdot \mathbf {a} =\left\|\mathbf {a} \right\|^{2},} which gives ‖ a ‖ = a ⋅ a , {\displaystyle \left\|\mathbf {a} \right\|={\sqrt {\mathbf {a} \cdot \mathbf {a} }},}

6120-709: The dot product of the vector [ 1 , 3 , − 5 ] {\displaystyle [1,3,-5]} with itself is:   [ 1 , 3 , − 5 ] ⋅ [ 1 , 3 , − 5 ] = ( 1 × 1 ) + ( 3 × 3 ) + ( − 5 × − 5 ) = 1 + 9 + 25 = 35 {\displaystyle {\begin{aligned}\ [1,3,-5]\cdot [1,3,-5]&=(1\times 1)+(3\times 3)+(-5\times -5)\\&=1+9+25\\&=35\end{aligned}}} If vectors are identified with column vectors ,

6210-522: The dot product takes two vectors and returns a scalar quantity. It is also known as the "scalar product". The dot product of two vectors can be defined as the product of the magnitudes of the two vectors and the cosine of the angle between the two vectors. Thus, a ⋅ b = | a | | b | cos ⁡ θ {\displaystyle \mathbf {a} \cdot \mathbf {b} =|\mathbf {a} |\,|\mathbf {b} |\cos \theta } Alternatively, it

6300-793: The field of Fourier analysis . Differentiating any sinusoid with respect to time can be viewed as multiplying its amplitude by its angular frequency and advancing it by a quarter cycle: d d t [ A sin ⁡ ( ω t + φ ) ] = A ω cos ⁡ ( ω t + φ ) = A ω sin ⁡ ( ω t + φ + π 2 ) . {\displaystyle {\begin{aligned}{\frac {d}{dt}}[A\sin(\omega t+\varphi )]&=A\omega \cos(\omega t+\varphi )\\&=A\omega \sin(\omega t+\varphi +{\tfrac {\pi }{2}})\,.\end{aligned}}} A differentiator has

6390-492: 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. 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 {\displaystyle n} vector u {\displaystyle u} is, then,

6480-402: The form: Since sine waves propagate without changing form in distributed linear systems , they are often used to analyze wave propagation . When two waves with the same amplitude and frequency traveling in opposite directions superpose each other, then a standing wave pattern is created. On a plucked string, the superimposing waves are the waves reflected from the fixed endpoints of

6570-537: The formula for the Euclidean length of the vector. The scalar projection (or scalar component) of a Euclidean vector a {\displaystyle \mathbf {a} } in the direction of a Euclidean vector b {\displaystyle \mathbf {b} } is given by a b = ‖ a ‖ cos ⁡ θ , {\displaystyle a_{b}=\left\|\mathbf {a} \right\|\cos \theta ,} where θ {\displaystyle \theta }

6660-421: The frequency of the strobe equals the frequency of the rotating or vibrating object, the object completes one cycle of oscillation and returns to its original position between the flashes of light, so when illuminated by the strobe the object appears stationary. Then the frequency can be read from the calibrated readout on the stroboscope. A downside of this method is that an object rotating at an integer multiple of

6750-414: The general form: y ( t ) = A sin ⁡ ( ω t + φ ) = A sin ⁡ ( 2 π f t + φ ) {\displaystyle y(t)=A\sin(\omega t+\varphi )=A\sin(2\pi ft+\varphi )} where: Sinusoids that exist in both position and time also have: Depending on their direction of travel, they can take

6840-633: The inner product on vectors uses a sum over corresponding components, the inner product on functions is defined as an integral over some measure space ( X , A , μ ) {\displaystyle (X,{\mathcal {A}},\mu )} : ⟨ u , v ⟩ = ∫ X u v d μ . {\displaystyle \left\langle u,v\right\rangle =\int _{X}uv\,{\text{d}}\mu .} For example, if f {\displaystyle f} and g {\displaystyle g} are continuous functions over

6930-413: The notions of Hermitian forms and general inner product spaces , which are widely used in mathematics and physics . The self dot product of a complex vector a ⋅ a = a H a {\displaystyle \mathbf {a} \cdot \mathbf {a} =\mathbf {a} ^{\mathsf {H}}\mathbf {a} } , involving the conjugate transpose of a row vector, is also known as

7020-545: The number of counts is not very large, it is more accurate to measure the time interval for a predetermined number of occurrences, rather than the number of occurrences within a specified time. The latter method introduces a random error into the count of between zero and one count, so on average half a count. This is called gating error and causes an average error in the calculated frequency of Δ f = 1 2 T m {\textstyle \Delta f={\frac {1}{2T_{\text{m}}}}} , or

7110-412: The points of space are defined in terms of their Cartesian coordinates , and Euclidean space itself is commonly identified with the real coordinate space R n {\displaystyle \mathbf {R} ^{n}} . In such a presentation, the notions of length and angle are defined by means of the dot product. The length of a vector is defined as the square root of the dot product of

7200-410: The reference frequency. To convert higher frequencies, several stages of heterodyning can be used. Current research is extending this method to infrared and light frequencies ( optical heterodyne detection ). Visible light is an electromagnetic wave , consisting of oscillating electric and magnetic fields traveling through space. The frequency of the wave determines its color: 400 THz ( 4 × 10 Hz)

7290-427: The result is a scalar , rather than a vector (as with the vector product in three-dimensional space). The dot product may be defined 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 ,

7380-560: The rotation rate of a shaft, mechanical vibrations, or sound waves , can be converted to a repetitive electronic signal by transducers and the signal applied to a frequency counter. As of 2018, frequency counters can cover the range up to about 100 GHz. This represents the limit of direct counting methods; frequencies above this must be measured by indirect methods. Above the range of frequency counters, frequencies of electromagnetic signals are often measured indirectly utilizing heterodyning ( frequency conversion ). A reference signal of

7470-414: The same speed (the speed of light), giving them wavelengths inversely proportional to their frequencies. c = f λ , {\displaystyle \displaystyle c=f\lambda ,} where c is the speed of light ( c in vacuum or less in other media), f is the frequency and λ is the wavelength. In dispersive media , such as glass, the speed depends somewhat on frequency, so

7560-500: The string. The string's resonant frequencies are the string's only possible standing waves, which only occur for wavelengths that are twice the string's length (corresponding to the fundamental frequency ) and integer divisions of that (corresponding to higher harmonics). The earlier equation gives the displacement y {\displaystyle y} of the wave at a position x {\displaystyle x} at time t {\displaystyle t} along

7650-461: The strobing frequency will also appear stationary. Higher frequencies are usually measured with a frequency counter . This is an electronic instrument which measures the frequency of an applied repetitive electronic signal and displays the result in hertz on a digital display . It uses digital logic to count the number of cycles during a time interval established by a precision quartz time base. Cyclic processes that are not electrical, such as

7740-435: The vector a = [ 1   i ] {\displaystyle \mathbf {a} =[1\ i]} ). 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 dot product, through the alternative definition a ⋅ b = ∑ i

7830-416: The vector by itself, and the cosine of the (non oriented) angle between 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. The dot product of two vectors a = [ a 1 , a 2 , ⋯ ,

7920-473: The wave speed is independent of frequency), frequency has an inverse relationship to the wavelength , λ ( lambda ). Even in dispersive media, the frequency f of a sinusoidal wave is equal to the phase velocity v of the wave divided by the wavelength λ of the wave: f = v λ . {\displaystyle f={\frac {v}{\lambda }}.} In the special case of electromagnetic waves in vacuum , then v = c , where c

8010-419: The wavelength is not quite inversely proportional to frequency. Sound propagates as mechanical vibration waves of pressure and displacement, in air or other substances. In general, frequency components of a sound determine its "color", its timbre . When speaking about the frequency (in singular) of a sound, it means the property that most determines its pitch . The frequencies an ear can hear are limited to

8100-462: The wavelength of the sound waves (distance between repetitions) is approximately inversely proportional to frequency. In Europe , Africa , Australia , southern South America , most of Asia , and Russia , the frequency of the alternating current in household electrical outlets is 50 Hz (close to the tone G), whereas in North America and northern South America, the frequency of

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