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Topic: Dot product

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Cross product

Inner product space

Net force

Evaluating cross products

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	Dot product - Wikipedia, the free encyclopedia
		In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity.
		As the cosine of 90° is zero, the dot product of two perpendicular vectors is always zero.
		The inner product generalizes the dot product to abstract vector spaces, it is normally denoted by .
	en.wikipedia.org /wiki/Dot_product (996 words)

	Product (mathematics) - Wikipedia, the free encyclopedia
		In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied.
		When matrices or members of various other associative algebras are multiplied the product usually depends on the order of the factors; in other words, matrix multiplication, and the multiplications in those other algebras, are non-commutative.
		The dot product and cross product are forms of multiplication of vectors; the same as dot product or more general are the scalar product and the inner product; see also Inner product space
	en.wikipedia.org /wiki/Product_(mathematics) (186 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.
		If one were to take the dot product of a unit vector A and a second vector B of any non-zero length, the result is the length of vector B projected in the direction of vector A (see illustration to left).
	www.mvps.org /directx/articles/math/dot (644 words)

	Function Dot in theory Tensor-Quantities
		The quantity.dimension of the product equals the product of the quantity.dimensions of the operands.
		The product is a dyad (second order tensor) and is equal to the sum of the products of the corresponding scalar components of the argument tensors for a common basis.
		The product is a vector-quantity (first order tensor) and is equal to the sum of the products of the corresponding scalar components of the argument tensors for a common basis.
	www-ksl.stanford.edu /knowledge-sharing/ontologies/html/tensor-quantities/DOT.html (195 words)

	Vector Products (Site not responding. Last check: 2007-11-07)
		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)

	dot product
		DOT PRODUCT (AKA inner product or scalar product)
		Another way to look at it is that the dot product is the magnitude of one vector times the component of the other in the direction of the first.
		Work done by a force is the dot product of the force acting on a body and the displacement of the body that occurs while the force acts.
	instruct.tri-c.edu /fgram/web/dotprod.htm (257 words)

	Development of the idea of the Vector Cross Product
		The dot product gives the vector amount that one vector contributes along the same line as another vector, and that straightforward result may serve as sufficient explanation for the dot 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.
	www.rtis.com /nat/user/jfullerton/school/math251/cproduct.htm (1138 words)

	Dot Product (Site not responding. Last check: 2007-11-07)
		The Dot Product applet shows how the scalar dot product value of two vectors depends on both the vectors' lengths and the angle between them.
		The dot product is an essential building block in linear algebra and for doing almost any type of transformation or rendering in computer graphics.
		The dot product value is shown with a thermometer-style display, as well as in an equation format, along with the vector coordinates.
	www.cs.brown.edu /exploratories/freeSoftware/repository/edu/brown/cs/exploratories/applets/dotProduct/dot_product_guide.html (162 words)

	World Web Math: Vector Calculus: Dot Product (Site not responding. Last check: 2007-11-07)
		The dot product is a miracle composed of two definitions.
		">d is equal to the product of the magnitude of the displacement and the component of the force in the direction of the displacement:
		Calculate the dot product of two unit vectors separated by an angle of 60 degrees.
	web.mit.edu /wwmath/vectorc/3d/dotp.html (647 words)

	The dot product
		The dot product of a and b, written a.
		One of the main uses of the dot product is to determine whether two vectors, a and b, are othogonal (perpendicular).
		Using the dot product we can find the angle subtended by our two position vectors, multiply by the radius of the Earth, and hey presto we have the great circle distance.
	members.tripod.com /~Paul_Kirby/vector/Vdotproduct.html (488 words)

	Define dot product - a definition from Whatis.com
		The dot product, also called the scalar product, of two vectors is a number (scalar quantity) obtained by performing a specific operation on the vector components.
		The dot product of two vectors is determined by multiplying their x-coordinates, then multiplying their y-coordinates, and finally adding the two products.
		When expressed in this format, the dot product of two vectors is equal to the product of their lengths, multiplied by the cosine of the angle between them.
	whatis.techtarget.com /definition/0,,sid9_gci802995,00.html (284 words)

	Vectors, Part 2
		Thus, the length of a vector is the square root of its dot product with itself.
		Here, the white vector is dotted with the green vector, and the corresponding component of the green vector in the direction of the white is shown in red when you press the "projection" button.
		Prove your statement for the dot product of a general vector (a,b) with either i or j.
	www.math.duke.edu /education/ccp/materials/mvcalc/vectors/vec1a.html (467 words)

	Dot Product (Site not responding. Last check: 2007-11-07)
		This java applet demonstrates the dot product, which is an important concept in linear algebra and physics.
		The projection of A onto B is shown in yellow, and the angle between the two is shown in orange.
		At the bottom of the screen are four bars which show the magnitude of four quantities: the length of A (red), the length of B (blue), the length of the projection of A onto B (yellow), and the dot product of A and B (green).
	www.falstad.com /dotproduct (163 words)

	Product Key Finder - CD Key Finder - by Insaneware
		You cannot find the product key (CD Key) on your own, as it appears to be gobbledygook to the human eye.
		And, we wish that we could refer you to a product that can retrieve installation codes from a compact disk, but such a product does not exist.
		You give the customer service agent the product ID and he or she will then give you your Office 2000 Product Key.
	www.product-key.com (909 words)

	PlanetMath: dot product (Site not responding. Last check: 2007-11-07)
		Then we define the dot product of the two vectors as:
		of scalar product is the scalar square of the vector
		This is version 8 of dot product, born on 2001-10-15, modified 2006-03-10.
	planetmath.org /encyclopedia/DotProduct.html (105 words)

	What exactly is the Dot Product? - GameDev.Net Discussion Forums
		The dot product is one way of multiplying vectors together to give a scalar, and you can't differentiate that meaning from all of the other ones we gave (they're all essentially equivalent, given the constraints that have been mentioned).
		The dot product is a fundamentally different operation and should use a different operator (or explicit function).
		Hence the dot product is defined as A.B = AB * cos(c) where c is the angle between A and B. And the rule for calculating A.B is (Ax * Bx + Ay * By).
	www.gamedev.net /community/forums/topic.asp?topic_id=285380 (2016 words)

	PHY 206 Review of Dot Product / Dr. Miner (Site not responding. Last check: 2007-11-07)
		It is to use the Scalar Product, or the Dot Product.
		Thus the result of Dot Product, or Scalar Product is simply a scalar.
		This simplifies the process since we note that the dot product of a unit vector with itself is equal to (1)(1)cos(0) = "1," whereas the dot product of two different unit vectors is always zero.
	www.udayton.edu /~physics/gkm/rdot.htm (247 words)

	Calculus II (Math 2414) - Vectors - Dot Product (Site not responding. Last check: 2007-11-07)
		The dot product gives us a very nice method for determining if two vectors are perpendicular and it will give another method for determining when two vectors are parallel. Note as well that often we will use the term orthogonal in place of perpendicular.
		There are several nice applications of the dot product as well that we should look at.
		This application of the dot product requires that we be in three dimensional space unlike all the other applications we’ve looked at to this point.
	tutorial.math.lamar.edu /AllBrowsers/2414/DotProduct.asp (665 words)

	SparkNotes: Vector Multiplication: The Dot Product
		Technically speaking, the dot product is a kind of scalar product.
		This equation is exactly the right formula for the dot product of two 3-dimensional vectors.
		In other words, the dot product of v with i picks off the component of v in the x-direction, and similarly v's dot product with j picks off the component of v which lies in the y-direction.
	www.sparknotes.com /physics/vectors/vectormultiplication/section1.html (961 words)

	Vector Dot Products
		Applications of the dot product will be shown.
		, is the product of the force and the distance through which the force is applied.
		A dot product can be used to calculate the angle between two vectors.
	www.algebralab.org /lessons/lesson.aspx?file=Trigonometry_TrigVectorDotProd.xml (358 words)

	Chapter 1 : Vector Dot Product (Site not responding. Last check: 2007-11-07)
		The Scalar product of two vector is defined as the product of their magnitudes multiplied by the cosine of the angle between the vectors.
		Thus the Scalar product of vectorA and vector B is written
		The dot product of two vectors yields a scalar.
	www.ee.surrey.ac.uk /Teaching/Courses/EFT/statics/html/chapter1d.html (80 words)

	Linear Algebra (Part 1.1) ~ 3DSoftware.com (Site not responding. Last check: 2007-11-07)
		The Scalar Product is produced by processing two vectors that have the same number of components (cells).
		In linear algebra, the terms scalar product, inner product and dot product mean the same thing, and are used interchangeably.
		The term inner product is particularly descriptive in our discussion here, since the multiplication is among internal parts (components) of the vectors, and –; as we shall see later – works inside matrices.
	www.3dsoftware.com /Math/Programming/LinAlg01 (387 words)

	Review of vector Mathematics (Site not responding. Last check: 2007-11-07)
		The dot product of two vectors is a scalar, so the dot product is sometimes called the `scalar product.' The dot product of two vectors
		In dynamics the dot product is used to define work and power, to reduce a vector to components, and to reduce vector equations to scalar equations.
		is a vector, the cross product is also called the vector product to distinguish it from the scalar product (the dot product).
	www.eng.fsu.edu /~ecollins/dynamics/vectors (347 words)

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