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Vectors

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4 Mar 2010 11:20:47 IST

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why vectors cant be divided...???

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why vectors cant be divided...???

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illuminatti

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Joined: 19 Feb 2010

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4 Mar 2010 14:46:28 IST

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Division of vectors does not yield a definite answer. This is due to the
fact that there are two kinds of multiplication that can produce a vector.

One option is multiplying a vector by a scalar. An example is multiplying a
velocity vector by the number 3. This produces a vector parallel to the
first vector: 3(vectorA)

The other kind of multiplication is the "cross-product", multiply one vector
by another to produce a vector that is perpendicular to both. One
difficulty involves parallel components. Consider:
(vectorA)x(vectorB)=(vectorC).
You can add vector B to vector A and still get the same product:
(vectorA + vectorB)x(vectorB)
=(vectorA)x(vectorB)+(vectorB)x(vectorB)=(vectorC)+0

The first process produces a product parallel to vectorA. The second
produces a product perpendicular to vectorA. And then there are all the
vectors that are neither parallel nor perpendicular to vector A.

Rather than division, the correct approach is to contemplate what
multiplication could have produced the product, and what restrictions apply.
This is more like saying "What can I multiply by seven to yield 42?", rather
than "What is 42 divided by 7?".

illuminatti

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Posts: 260

4 Mar 2010 14:51:16 IST

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geometric algebra has a well defined, and very useful, vector multiplication and divison operation (geometric product, or clifford product). It's not closed, nor generally commutative, and produces a scalar and bivector (wedge) product. An example application of division is orthogonal decomposition with respect to a reference vector:

$LaTeX Code: <BR>x = \\frac{1}{a} a x = \\frac{a}{a^2} a x = \\frac{a}{a^2} (a \\cdot x + a \\wedge x) = \\hat{a} (\\hat{a} \\cdot x) + \\hat{a} (\\hat{a} \\wedge x)<BR>$

This first part is the familiar vector projection operation, and the second is the orthogonal component.

The basic rules for the multiplication(/division), assuming euclianian metric are:

- associativity
- linearity
- $LaTeX Code: a^2 = {\\lvert a \\rvert}^2$

As a subject this ties together quite an astounding number of seemingly unrelated bits and pieces (complex numbers and quaternions, dot & cross product, linear equation solution, and much more.)

edison

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Joined: 19 Oct 2006

Posts: 7449

4 Mar 2010 17:10:21 IST

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Vector Division

In general, there is no unique matrix solution to the matrix equation

Even in the case of parallel to , there are still multiple matrices that perform this transformation. For example, given , all the following matrices satisfy the above equation:

Therefore, vector division cannot be uniquely defined in terms of matrices.

However, if the vectors are represented by complex numbers or quaternions, vector division can be uniquely defined using the usual rules of complex division and quaternion algebra, respectively.

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