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Dot product

In mathematics, the dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number obtained by multiplying corresponding entries and adding up those products. The name is derived from the centered dot "·" that is often used to designate this operation; the alternative name scalar product emphasizes the scalar (rather than vector) nature of the result.

The principal use of this product is the inner productin aEuclidean vector space: when two vectors are expressed on an orthonormal basis, the dot product of their coordinate vectors gives their inner product. For this geometric interpretation, scalars must be taken to be real numbers; while the dot product can be defined in a more general setting (for instance with complex numbers as scalars) many properties would be different. The dot product contrasts (in three dimensional space) with the cross product, which produces a vector as result.

Definition

The dot product of two vectors a = [a1, a2, ... , an] and b = [b1, b2, ... , bn] is defined as:

\mathbf{a}\cdot \mathbf{b} = \sum_{i=1}^n a_ib_i = a_1b_1 + a_2b_2 + \cdots + a_nb_n

where Σ denotes summation notation and n is the dimension of the vector space.

In dimension 2, the dot product of vectors [a,b] and [c,d] is ac + bd. Similarly, in a dimension 3, the dot product of vectors [a,b,c] and [d,e,f] is ad + be + cf. For example, the dot product of two three-dimensional vectors [1, 3, −5] and [4, −2, −1] is

[1, 3, -5] \cdot [4, -2, -1]

The dot product can also be obtained via transposition and matrix multiplication as follows:

\mathbf{b}^\mathrm{T}\mathbf{a} = \mathbf{a}^\mathrm{T}\mathbf{b},

where both vectors are interpreted as column vectors, and aT denotes the transpose of a, in other words the corresponding row vector.

Geometric interpretation

In Euclidean geometry, the dot product, length, and angle are related. For a vector a, the dot product a· a is the square of the length of a, or

{\mathbf{a} \cdot \mathbf{a}}=\left\|\mathbf{a}\right\|^2 = |\mathbf{a}|^2

where ||a|| (also written |a|) denotes the length (magnitude) of a. More generally, if b is another vector

\mathbf{a} \cdot \mathbf{b}=\left\|\mathbf{a}\right\| \, \left\|\mathbf{b}\right\| \cos \theta \,

where ||a|| and ||b|| denote the length of a and b and θ is the angle between them.

This formula can be rearranged to determine the size of the angle between two nonzero vectors:

\theta=\arccos \left( \frac {\bold{a}\cdot\bold{b}} {\left\|\bold{a}\right\|\left\|\bold{b}\right\|}\right)

One can also first convert the vectors to unit vectors by dividing by their magnitude:

\boldsymbol{\hat{a}} = \frac{\bold{a}}{\left\|\bold{a}\right\|}

then the angle θ is given by

\theta = \arccos ( \boldsymbol{\hat a}\cdot\boldsymbol{\hat b})

The terminal points of both unit vectors lie on the unit circle. The unit circle is where the trigonometric values for the six trig functions are found. After substitution, the first vector component is cosine and the second vector component is sine, i.e. (cos x, sin x) for some angle x. The dot product of the two unit vectors then takes x, sin x>y, sin y> for angles x, y and returns (cos x)(cos y) + (sin x)(sin y) = cos(x− y) where x− y = theta.

As the cosine of 90° is zero, the dot product of two orthogonal vectors is always zero. Moreover, two vectors can be considered orthogonal if and only if their dot product is zero, and they have non-null length. This property provides a simple method to test the condition of orthogonality.

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.

The geometric properties rely on the basis being orthonormal, i.e. composed of pairwise perpendicular vectors with unit length.

Scalar projection

If both a and b have length one (i.e., they are unit vectors), their dot product simply gives the cosine of the angle between them.

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 projection of a onto b, or 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).

If neither a nor b is a unit vector, then the magnitude of the projection of a in the direction of b is a· (b / |b|), as the unit vector in the direction of b is b / |b|.

Rotation

A rotation of the orthonormal basis in terms of which vector a is represented is obtained with a multiplication of a b

Cross product

In mathematics, the cross product, vector product, or Gibbs vector product is abinary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them. It has many applications in mathematics, engineering and physics.

If either of the vectors being multiplied is zero or the vectors are parallel then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular for perpendicular vectors this is a rectangle and the magnitude of the product is the product of their lengths. The cross product is anticommutative, distributive over addition and satisfies the Jacobi identity. The space and product form an algebra over a field, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.

Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on the choice of orientation or "handedness". The product can be generalized in various ways; it can be made independent of orientation by changing the result to pseudovector, or in arbitrary dimensions the exterior product of vectors can be used with a bivector or two-form result. Also, using the orientation and metric structure just as for the traditional 3d cross product, one can in n dimensions take the product of n - 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results it only exists in three and seven dimensions.

Definition

The cross product of two vectors a and b is denoted by In physics, sometimes the notation a∧b is used, though this is avoided in mathematics to avoid confusion with the exterior product.

The cross product a× b is defined as a vector c that is perpendicular to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.

The cross product is defined by the formula

\mathbf{a} \times \mathbf{b} = a b \sin \theta \ \mathbf{n}

where θ is the measure of the smaller angle between a and b (0° ≤ θ≤ 180°), a and b are the magnitudes of vectors a and b, and n is a unit vectorperpendicular to the plane containing a and b in the direction given by the right-hand rule as illustrated. If the vectors a and b are parallel (i.e., the angle θ between them is either 0° or 180°), by the above formula, the cross product of a and b is the zero vector 0.

The direction of the vector n is given by the right-hand rule, where one simply points the forefinger of the right hand in the direction of a and the middle finger in the direction of b. Then, the vector n is coming out of the thumb (see the picture on the right). Using this rule implies that the cross-product is anti-commutative, i.e., b× a = -(a× b). By pointing the forefinger toward b first, and then pointing the middle finger toward a, the thumb will be forced in the opposite direction, reversing the sign of the product vector.

Using the cross product requires the handedness of the coordinate system to be taken into account (as explicit in the definition above). If a left-handed coordinate system is used, the direction of the vector n is given by the left-hand rule and points in the opposite direction.

This, however, creates a problem because transforming from one arbitrary reference system to another (e.g., a mirror image transformation from a right-handed to a left-handed coordinate system), should not change the direction of n. The problem is clarified by realizing that the cross-product of two vectors is not a (true) vector, but rather a pseudovector. Seecross product and handedness for more detail.

Computing the cross product

Coordinate notation

The unit vectors i, j, and k from the given orthogonal coordinate system satisfy the following equalities:

i× j = k          j× k = i          k× i = j

Together with the skew-symmetry and bilinearity of the cross product, these three identities are sufficient to determine the cross product of any two vectors. In particular, the following identities are also seen to hold

j× i = −k    &

Depleted uranium

Depleted uranium (DU) is uranium primarily composed of the isotopeuranium-238 (U-238). Natural uranium is about 99.27 percent U-238, 0.72 percent U-235, and 0.0055 percent U-234. U-235 is used for fission in nuclear reactors and nuclear weapons. Uranium is enriched in U-235 by separating the isotopes by mass. The byproduct of enrichment, called depleted uranium or DU, contains less than one third as much U-235 and U-234 as natural uranium. The external radiation dose from DU is about 60 percent of that from the same mass of natural uranium. DU is also found in reprocessed spent nuclear reactor fuel, but that kind can be distinguished from DU produced as a byproduct of uranium enrichment by the presence of U-236. In the past, DU has been called Q-metal, depletalloy, and D-38.

DU is useful because of its very high density of 19.1 g/cm3 (68.4% denser than lead). Civilian uses include counterweights in aircraft, radiation shielding in medical radiation therapy and industrial radiography equipment, and containers used to transport radioactive materials. Military uses include defensive armor plating and armor-piercingprojectiles.

The use of DU in munitions is controversial because of questions about potential long-term health effects. Normal functioning of the kidney, brain, liver, heart, and numerous other systems can be affected by uranium exposure, because uranium is a toxic metal. It is weakly radioactive and remains so because of its long physical half-life (4.468 billion years for uranium-238). The biological half-life (the average time it takes for the human body to eliminate half the amount in the body) for uranium is about 15 days. The aerosol produced during impact and combustion of depleted uranium munitions can potentially contaminate wide areas around the impact sites leading to possible inhalation by human beings. During a three week period of conflict in 2003 in Iraq, 1,000 to 2,000 tonnes of DU munitions were used.

The actual acute and chronic toxicity of DU is also a point of medical controversy. Multiple studies using cultured cells and laboratory rodents suggest the possibility of leukemogenic, genetic, reproductive, and neurological effects from chronic exposure. A 2005 epidemiology review concluded: "In aggregate the human epidemiological evidence is consistent with increased risk of birth defects in offspring of persons exposed to DU." The World Health Organization, the directing and coordinating authority for health within the United Nations which is responsible for setting health research norms and standards, providing technical support to countries and monitoring and assessing health trends, states that no risk of reproductive, developmental, or carcinogenic effects have been reported in humans due to DU exposure. This report has been criticized by Dr. Keith Baverstock for not including possible long term effects of DU on human body.

History

Enriched uranium was first manufactured in the 1940s when the US, UK, France and USSR began their nuclear weapons and nuclear power programs. It was at this time that depleted uranium was first stored as an unusable waste product (uranium hexafluoride). There was some hope that the enrichment process would be improved and fissionable isotopes of U-235 could, at some future date, be extracted from the depleted uranium. This re-enrichment recovery of the residual uranium-235 contained in the depleted uranium is no longer a matter of the future: it has been practiced for several years. Also, it is possible to design civilian power reactors with unenriched fuel, but only about 10 percent of reactors ever built utilize that technology, and both nuclear weapons production and naval reactors require the concentrated isotope.

In the 1970s, the Pentagon reported that the Soviet military had developed armor plating for Warsaw Pact tanks that NATO ammunition could not penetrate. The Pentagon began searching for material to make denser bullets. After testing various metals, ordnance researchers settled on depleted uranium.

The US

Unit vector

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length). A unit vector is often denoted by a lowercase letter with a “hat�, like this: {\hat{\imath}} (pronounced "i-hat").

In Euclidean space, the dot product of two unit vectors is simply the cosine of the angle between them. This follows from the formula for the dot product, since the lengths are both 1.

The normalized vector or versor \boldsymbol{\hat{u}} of a non-zero vector \boldsymbol{u} is the unit vector codirectional with \boldsymbol{u}, i.e.,

\boldsymbol{\hat{u}} = \frac{\boldsymbol{u}}{\|\boldsymbol{u}\|}

where \|\boldsymbol{u}\| is the norm (or length) of \boldsymbol{u}. The term normalized vector is sometimes used as a synonym for unit vector.

The elements of a basis are usually chosen to be unit vectors. Every vector in the space may be written as a linear combination of unit vectors. The most commonly encountered bases are Cartesian, polar, and spherical coordinates. Each uses different unit vectors according to the symmetry of the coordinate system. Since these systems are encountered in so many different contexts, it is not uncommon to encounter different naming conventions than those used here.

Cartesian coordinates

In the three dimensional Cartesian coordinate system, the unit vectors codirectional with the x, y, and z axes are sometimes referred to as versors of the coordinate system.

\mathbf{\hat{\boldsymbol{\imath}}} = \begin{bmatrix}1\\0\\0\end{bmatrix}, \,\, \mathbf{\hat{\boldsymbol{\jmath}}} = \begin{bmatrix}0\\1\\0\end{bmatrix}, \,\, \mathbf{\hat{\boldsymbol{k}}} = \begin{bmatrix}0\\0\\1\end{bmatrix}

These are often written using normal vector notation (e.g. i, or \vec{\imath}) rather than the caret notation, and in most contexts it can be assumed that i, j, and k, (or \vec{\imath}, \vec{\jmath}, and \vec{k}) are versors of a Cartesian coordinate system (hence a tern of mutually orthogonal unit vectors). The notations (\boldsymbol\hat{x}, \boldsymbol\hat{y}, \boldsymbol\hat{z}), (\boldsymbol\hat{x}_1, \boldsymbol\hat{x}_2, \boldsymbol\hat{x}_3), (\boldsymbol\hat{e}_x, \boldsymbol\hat{e}_y, \boldsymbol\hat{e}_z), or (\boldsymbol\hat{e}_1, \boldsymbol\hat{e}_2, \boldsymbol\hat{e}_3), with or without hat/caret, are also used, particularly in contexts where i, j, k might lead to confusion with another quantity (for instance with index symbols such as i, j, k, used to identify an element of a set or array or sequence of variables). These vectors represent an example of a standard basis.

When a unit vector in space is expressed, with Cartesian notation, as a linear combination of i, j, k, its three scalar components can be referred to as direction cosines. The value of each component is equal to the cosine of the angle formed by the unit vector with the respective basis vector. This is one of the methods used to describe the orientation (angular position) of a straight line, segment of straight line, oriented axis, or segment of oriented axis (vector).

Cylindrical coordinates

The unit vectors appropriate to cylindrical symmetry are: \boldsymbol{\hat{s}} (also designated \boldsymbol{\hat{r}} or \boldsymbol{\hat \rho}), the distance from the axis of symmetry; \boldsymbol{\hat \varphi}, the angle measured counterclockwise from the positive x-axis; and \boldsymbol{\hat{z}}. They are related to the Cartesian basis \hat{x}, \hat{y}, \hat{z} by:

\boldsymbol{\hat{s}} = \cos \varphi\boldsymbol{\hat{x}} + \sin \varphi\boldsymbol{\hat{y}}
\boldsymbol{\hat \varphi} = -\sin \varphi\boldsymbol{\hat{x}} + \cos \varphi\boldsymbol{\hat{y}}
\boldsymbol{\hat{z}}=\boldsymbol{\hat{z}}.

It is important to note that \boldsymbol{\hat{s}} and \boldsymbol{\hat \varphi} are functions of \varphi, and are not constant in direction. When differentiating or integrating in cylindrical coordinates, these unit vectors themselves must also be operated on. For a more complete description, see Jacobian. The derivatives with respect to \varphi are:

\frac{\partial \boldsymbol{\hat{s}}} {\partial \varphi} = -\sin \varphi\boldsymbol{\hat{x}} + \cos \varphi\boldsymbol{\hat{y}} = \boldsymbol{\hat \varphi}
\frac{\partial \boldsymbol{\hat \varphi}} {\partial \varphi} = -\cos \varphi\boldsymbol{\hat{x}} - \sin \varphi\boldsymbol{\hat{y}} = -\boldsymbol{\hat{s}}
\frac{\partial \boldsymbol{\hat{z}}} {\partial \varphi} = \mathbf{0}.

Spherical coordinates

The unit vectors appropriate to spherical symmetry are: \boldsymbol{\hat{r}}, the radial distance from the origin; \boldsymbol{\hat{\varphi}}, the angle in the x-y plane counterclockwise from the positive x-axis; and \boldsymbol{\hat \theta}, the angle from the positive z axis. To minimize degeneracy, the polar angle is usually taken 0\leq\theta\leq 180^\circ. It is especially important to note the context of any ordered triplet written in spherical coordinates, as the roles of \boldsymbol{\hat \varphi} and \boldsymbol{\hat \theta} are often reversed. Here, the American naming convention is used. This leaves the azimuthal angle \varphi defined the same as in cylindrical coordinates. The Cartesian relations are:

\boldsymbol{\hat{r}} = \sin \theta \cos \varphi\boldsymbol{\hat{x}} + \sin \theta \sin \varphi\boldsymbol{\hat{y}} + \cos \theta\boldsymbol{\hat{z}}
\boldsymbol{\hat \theta} = \cos \theta \cos \varphi\boldsymbol{\hat{x}} + \cos \theta \sin \varphi\boldsymbol{\hat{y}} - \sin \theta\boldsymbol{\hat{z}}
\boldsymbol{\hat \varphi} = - \sin \varphi\boldsymbol{\hat{x}} + \cos \varphi\boldsymbol{\hat{y}}

The spherical unit vectors depend on both \varphi and \thet


From Yahoo Answers

Question:I am doing E&M and need to do a cross product in spherical. I am pretty sure I can't just use a straight determinant like in rectangular coordinates, but I can't remember what changes in a spherical determinant. I can't find the definition of it in my books or online. Appreciate the help.

Answers:It is a bit different, though the same basic concept. Let's say you have two coordinates, (A, B, C) and (X, Y, Z). Your cross product will be (BZ - CY, XC - ZA, AY - BX). Give me a second to type how to remember it: First, position the coordinates above each other visually, with the first one on top (yes, order does matter): (A, B, C) (X, Y, Z) For the first term, cover up the first values: ( , B, C) ( , Y, Z) and then multiply the B times Z, subtracting from that C times Y (BZ - CY). For the second term, cover the middle values: (A, , C) (X, , Z) And multiply and subtract again, but this time switch directions. Start with the first value in the second coordinate (XC - ZA) For the third term, cover the last values: (A, B, ) (X, Y, ) and then multiply and subtract, starting with the value in the first coordinate again (AY - BX).

Question:How do you do it? is it just like Cartesian or is it different? (by different i mean just like when we perform the curl, in spherical and cylindrical we have to throw in a bunch of r's sin's and cosine's)

Answers:The formulas that involve derivatives (like curl) are different in different coordinate systems, because the unit vectors are in different directions at different points in space. You have to take that into account when you take the limits to find the derivatives. That's where the r's and trig functions come from. Formulas that don't involve derivatives (like vector cross product) are the same in any orthogonal coordinate system (like cartesian, cylindrical, or spherical), because they only involve to vectors at one point in space. If you defined a coordinate system where the unit vectors were not orthogonal, then the "derivative free" formulas would be different in that system.

Question:in 3d vector a * b and a x b can you give me an example too? for example suppose a = <1,4,2> and b = <5,2,1>

Answers:Let the coordinates (in standard reference triad) be: a = (ax, ay, az) b = (bx, by, bz) inner or dot product a * b = ax*bx +ay*by + az*bz note that the result of an inner product is a real number. cross product a \times b is a 3d vector, its component is (ay*bz-by*az, bx*az - ax*bz, ax*by - bx*ay) if two vectors a and b are normal to each other, their inner product is the real number 0; If two vectors a and b are parallel to each other, their cross product is the vector (0, 0, 0). Hope this helps.

Question:Consider an orthogonal transformation L from R^n to R^n. Show that L preserves the dot product: v*w = L(v) * L(w) for all v and w in R^n.

Answers:I'm not sure if this will help. It depends on what theorems you have at this point to work with. Since L in a linear transformation from R^n to itself, There exists a matrix A such that L(u) = Au for every u in R^n. Since L is orthogonal, the matrix A is orthogonal. That is, A^T = A^(-1). Now, the dot product on R^n can be expressed as a matrix product. = v^Tu. So consider the dot product of L(u) and L(v) for any u,v in R^n. = = (Av)^T(Au) = (v^T A^T)(Au) = v^T (A^TA)u But A^TA=I so the last expression simplifies to v^Tu. That is = v^Tu = . This argument assumes that you can call on the fact that a linear operator (linear transformation from a vector space back to itself) can always be equated with a matrix product for coordinate vectors. Since the space in question is R^n, there must exist A such that L(u)=Au. The columns of A are obtained by finding L(ei) for each basis vector ei. Next, it requires that you know that A is an orthogonal matrix. Finally, it requires that you are allowed to use the fact that the dot product can be represented by matrix multiplication.