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Topic: Vector product

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	Vector Product of Vectors
		The vector product and the scalar product are the two ways of multiplying vectors which see the most application in physics and astronomy.
		The vector product of A and B is always perpendicular to both A and B.
		Another way of stating that is to say that the vector product is perpendicular to the plane formed by vectors A and B. This right-hand rule direction is produced mathematically by the vector product expression.
	hyperphysics.phy-astr.gsu.edu /hbase/vvec.html (311 words)

	Vector Operations (Site not responding. Last check: 2007-10-10)
		In Cartesian coordinates, the dot product is numerically equal to the sum of the products of the vector components.
		The dot product does generalize to higher dimensions, and the dot product is still the product of the magnitudes times the cosine of the angle between the vectors, and it is still equal to the sum of the products of the components.
		The cross product of two vectors is a vector quantity with a magnitude equal to the product of the magnitudes of the two vectors times the sine of the angle between them.
	www.mathrec.org /vector.html (1630 words)

	PlanetMath: cross product
		is a vector orthogonal to the plane of the two vectors being crossed, whose magnitude is equal to the area of the parallelogram defined by the two vectors.
		The cross product produces the vector that would be in a right-handed coordinate system with the plane.
		This is version 42 of cross product, born on 2001-11-15, modified 2006-09-13.
	planetmath.org /encyclopedia/CrossProduct.html (332 words)

	PlanetPhysics: Vector Triple Product (Site not responding. Last check: 2007-10-10)
		The vector triple product is the vector product of two vectors of which one is itself a vector product.
		This product is termed the vector triple product in contrast to the scalar triple product.
		This geometric use of the product is valuable not only in itself but for the light it sheds upon the properties of the product.
	planetphysics.org /encyclopedia/VectorTripleProduct.html (380 words)

	Vector
		Vector A and vector B are equal if, and only if, their representative components are the same.
		The addition of two vectors is a vector whose components are sums of the components of the vectors.
		In this case, the work is the product of the distance moved (the magnitude of the displacement vector) and the magnitude of the component of the force that acts in the direction of displacement (the scalar projection of F onto d):
	www.bsu.edu /web/jkshim/mathandstat/vector/vector.htm (920 words)

	GameDev.net -- Vectors and Matrices: A Primer
		i is a unit vector aligned with the x axis, j is a unit vector aligned with the y axis, and k is a unit vector aligned with the z axis.
		The components of this vector are the i, j and k coefficients (2, 3 and 5).
		The vector c is the vector from the end of the second vector to the end of the first, which in this case is from the end of b to the end of a.
	www.gamedev.net /reference/programming/features/vecmatprimer (1734 words)

	Cross product - Wikipedia, the free encyclopedia
		It is defined as the vector which is perpendicular to both a and b with a magnitude equal to the area of the parallelogram they span.
		Which vector is the "correct" one by convention depends upon the orientation of the vector space—i.e., on the handedness of the given orthogonal coordinate system (i, j, k).
		In the context of multilinear algebra, it is possible to define a generalized cross product in terms of parity such that the generalized cross product between two vectors of dimension n is a tensor of rank n−2.
	en.wikipedia.org /wiki/Cross_product (1797 words)

	Vector Products
		In the most trivial case the dot product of two vectors that are in the same direction is just the product of the magnitudes of the two vectors.
		The magnitude of the cross product is equal to the area of a parallelogram formed using the vectors as the sides of a parallelogram.
		The direction of the cross product is perpendicular to the plane formed by the two vectors and follows the right hand rule.
	www.ac.wwu.edu /~vawter/PhysicsNet/Topics/Vectors/VectorProducts.html (202 words)

	PlanetMath: vector product in general vector spaces
		One can easily see that some of the properties of the vector product are the same as in
		"vector product in general vector spaces" is owned by mathwizard.
		This is version 2 of vector product in general vector spaces, born on 2004-08-09, modified 2004-08-09.
	planetmath.org /encyclopedia/VectorProductInGeneralVectorSpaces.html (250 words)

	Vector Algebra:
		Addition of two vectors is accomplished by laying the vectors head to tail in sequence to create a triangle such as is shown in the figure.
		The orthogonal projection of a vector along a line is obtained by moving one end of the vector onto the line and dropping a perpendicular onto the line from the other end of the vector.
		The cross product of vectors a and b is a vector perpendicular to both a and b and has a magnitude equal to the area of the parallelogram generated from a and b.
	em-ntserver.unl.edu /Math/mathweb/vectors/vectors.html (1182 words)

	Vector Arithmetic
		The product of a scalar times a vector is a vector whose components are the components of the original vector, each multiplied by the scalar.
		Remember that the sum of two vectors is a vector from the tail of the first to the head of the second.
		This may be thought of as the product of the length of one vector times the component of the other vector in the direction of the first.
	www.mcasco.com /p1va.html (1972 words)

	vector. The Columbia Encyclopedia, Sixth Edition. 2001-05
		The vector, or cross, product of A and B is a vector, A × B, whose magnitude is equal to \|A\| \|B\| sin
		The vector product is an example of a kind of multiplication that does not follow the commutative law, since A × B = -B × A. Vector Analysis and Vector Space
		The methods of the calculus may be applied to such vector functions, leading to the branch of mathematics known as vector analysis.
	www.bartleby.com /65/ve/vector.html (713 words)

	Vectors
		Graphically, a vector is represented by an arrow, defining the direction, and the length of the arrow defines the vector's magnitude.
		The sum of two vectors, A and B, is a vector C, which is obtained by placing the initial point of B on the final point of A, and then drawing a line from the initial point of A to the final point of B, as illustrated in Panel 4.
		The scalar product of two vectors, A and B denoted by A·B, is defined as the product of the magnitudes of the vectors times the cosine of the angle between them, as illustrated in Panel 16.
	www.physics.uoguelph.ca /tutorials/vectors/vectors.html (1708 words)

	Development of the idea of the Vector Cross Product
		The cross product, however, is partly the result of multiplying different components of two vectors to get a product vector that is at right angles to both of the original vectors and that has a magnitude equal to the area of the parallelogram that the two vectors frame.
		The magnitude of the cross product is equivalent to the area of the parallelogram that the two vectors (a and b) describe, and the formula for the vector product is the same as the determinant for the array shown here.
		It is true that this area (or the set of geometrically equal areas) determines a vector (or set of vectors) perpendicular to that area, and that the vector (or vectors) so defined is precisely that vector which is the product in the modern form of the multiplication.
	www.rtis.com /nat/user/jfullerton/school/math251/cproduct.htm (1138 words)

	Vector Multiplication
		Since the projection of a vector on to itself leaves its magnitude unchanged, the dot product of any vector with itself is the square of that vector's magnitude.
		The dot product of two vectors is thus the sum of the products of their parallel components.
		For those of you familiar with matrices, the cross product of two vectors is the determinant of the matrix whose first row is the unit vectors, second row is the first vector, and third row is the second vector.
	hypertextbook.com /physics/foundations/vector-multiplication (807 words)

	Vector product (Site not responding. Last check: 2007-10-10)
		The magnitude of the resulting vector is equal to the product of the magnitude of vector with the scalar quantity.
		Recall that “bcos θ” is the scalar component of vector b along the direction of vector a and “a cos θ” is the scalar component of vector a along the direction of vector b.
		Similarly, the scalar component of vector in figure (i) is obtained by drawing perpendicular from the tip of the vector, b, on the direction of vector, a.
	cnx.org /content/m13603/latest (1987 words)

	Fizzics Fizzle: Advanced: Vector Products
		Many times in physics we talk about the products of vectors, but with something that has a direction and a magnitude (two parts), you probably don't know how to multiply them unless you have had exposure to dot and cross Products.
		Vector Product: yields a vector with a direction perpendicular to the plane formed by the two vectors being multiplied; commonly called the cross product
		The cross product is a vector quantity with a direction perpendicular to the plane of the two vectors and a magnitude given by the above equation.
	library.thinkquest.org /16600/advanced/vectorproducts.shtml (481 words)

	Understanding the Dot Product
		A dot product is a scalar value that is the result of an operation of two vectors with the same number of components.
		In this case, the dot product is equal to the cosine of the angle between the vectors.
		Angles between non-unit vectors (vectors with lengths not equal to 1.0) can be calculated either by first normalizing the vectors, or by dividing the dot product of the non-unit vectors by the length of each vector.
	www.mvps.org /directx/articles/math/dot (644 words)

	Page 433 (Site not responding. Last check: 2007-10-10)
		Vectors can be multiplied in two different ways: the scalar and vector product.
		As the name says, a scalar product of two vectors results in a scalar quantity, and a vector product in a vector quantity.
		The vector product between two vector is denoted by a cross (the product is sometimes also called "cross-product"):
	lectureonline.cl.msu.edu /~mmp/kap17/cd433.htm (125 words)

	Linear Algebra (1.3) ~ 3DSoftware.com
		The cross product produces a third vector which is perpendicular to the two vectors being “multiplied.” The length of this vector product, we will show in the next article (you do not need to learn this now), is the area of the parallelogram bounded by the two original vectors.
		One way to do that is to specify the moment arm as a vector (distance displacement from the hinge to the point on the force line of action that is closest to the hinge), and specify the other vector as the force (coordinate displacement of the force magnitude and direction).
		The resulting vector is perpendicular to the two vectors being multiplied (crossed), and has a magnitude (length) equal to the moment of force (torque).
	www.3dsoftware.com /Math/Programming/LinAlg01/03 (1254 words)

	Vector Development: LEADTOOLS Vector Products Overview (Site not responding. Last check: 2007-10-10)
		LEADTOOLS Vector Imaging technology is a great solution for developers who need to develop CAD/CAM/CAE applications, applications in support of AutoDesk AutoCad versions 12, 13, 14 and 2000, or any application that requires the use of vector images.
		With LEADTOOLS Vector Imaging, 2-D and 3-D vector imaging for loading, viewing, modifying and saving vector files to memory/disk in native vector formats is a matter of a few lines of code.
		Vector images can also be converted and saved as any of the raster formats supported by LEAD.
	www.leadtools.com /SDK/Vector/Vector-Products-n.htm (455 words)

	The vector product
		The cross product transforms like a vector, which means that it must have a well-defined direction and magnitude.
		By definition, the vector area has the magnitude of the scalar area, and is normal to the plane of the parallelogram, which means that it is perpendicular to both
		is the product of the magnitude of the force and the length of the lever arm
	farside.ph.utexas.edu /teaching/336k/lectures/node10.html (252 words)

	Vector Cross Product - JAVA Interactive Tutorial (Site not responding. Last check: 2007-10-10)
		The vector c is calculated from the vectors a and b using the
		The vector c calculated using the cross-product rules is always perpendicular (or "normal") to this plane.
		The magnitude c of the cross product is given by the formula c = ab sin(\phi), where \phi is the angle included between a and b.
	www.phy.syr.edu /courses/java-suite/crosspro.html (474 words)

	3. VECTORS - Title
		Although the decomposition of a vector depends on the coordinate system chosen, relations between vectors are not affected by the choice of the coordinate system (for example, if two vectors are perpendicular in one coordinate system, they are perpendicular in every coordinate system).
		The magnitude of the new vector is the magnitude of
		It is clear from the definition of the vector product that the order of the components is important.
	teacher.nsrl.rochester.edu /phy121/LectureNotes/Chapter03/Chapter3.html (1012 words)

	Products of Vectors
		The components of the vector are multiplied by the scalar and the result is a scaled vector which in the same direction as the original vector if the scalar is positive, or in the opposite direction if the scalar is negative.
		B of two vectors A and B is not a vector, but a scalar quantity (a number with units).
		The scalar product of two vectors A and B is a scalar quantity equal to the product of the magnitudes of the two vectors and the cosine of the smallest angle between them.
	electron9.phys.utk.edu /vectors/product.htm (587 words)

	Vector Product Applet (Site not responding. Last check: 2007-10-10)
		The vector product (or "cross product") of two vectors A and B is defined as:
		With the sliders on the right, you can adjust the lengths of the vectors, r, as well as their angles relative to the z-axis, theta, and x-axis, phi.
		To give a more precise indication of the location of the vectors in 3-d, we show their projections into the xy plane as well.
	chair.pa.msu.edu /applets/vector/a.htm (105 words)

	Vector (spatial) - Wikipedia, the free encyclopedia
		The magnitude of the vector is 15 N in both cases.
		The magnitude of the vector is 4 m in both cases.
		This happens to be equal to the square root of the dot product of the vector with itself:
	en.wikipedia.org /wiki/Vector_(spatial) (3196 words)

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