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Valentin Fadeev

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Edited by Valentin Fadeev, Sunday, 23 Jan 2011, 21:49

Had to do some revision of vector calculus/analysis before embarking on M828.

One point which I was not really missing, but did not quite get to grips with was the double vector product. I remembered the formula:

equation left hand side a right arrow multiplication open b right arrow multiplication c right arrow close equals right hand side b right arrow times open a right arrow dot operator c right arrow close minus c right arrow times open a right arrow dot operator b right arrow close ,

but nevertheless had difficulties applying it in excercises.

The reason whas that that the proof I saw used the expression of vector product in coordinates and comparison of both sides of the equation. However, I was aware of another, purely "vector" argument with no reference to any coordinate system.

Eventually I was able to reproduce only part of it, consulting one old textbook for some special trick. So here's how it goes.

For b right arrow multiplication c right arrow is perpendicular to the plane of b right arrow and c right arrow , a right arrow multiplication open b right arrow multiplication c right arrow close must lie in this plane, therefore:

equation left hand side a right arrow multiplication open b right arrow multiplication c right arrow close equals right hand side lamda times b right arrow plus mu times c right arrow

Dot-multiply both parts by a right arrow :

equation left hand side open a right arrow multiplication open b right arrow multiplication c right arrow close dot operator a right arrow close equals right hand side lamda times open b right arrow dot operator a right arrow close plus mu times open c right arrow dot operator a right arrow close

Since a right arrow multiplication open b right arrow multiplication c right arrow close up tack a right arrow , left-hand side is 0, so:

lamda times open b right arrow dot operator a right arrow close plus mu times open c right arrow dot operator a right arrow close equals zero reverse solidus qquod open asterisk operator close

Now define vector c prime right arrow lying in the plane of b right arrow and c right arrow , perpendicular to c right arrow and directed so that c prime right arrow , c right arrow and b right arrow multiplication c right arrow form the left-hand oriented system. This guarantees that the angle between b right arrow and c prime right arrow , open times times b right arrow c prime right arrow hat close less than pi divided by two .

Dot-multiply both parts by c prime right arrow :

equation left hand side open a right arrow multiplication open b right arrow multiplication c right arrow close dot operator c prime right arrow close equals right hand side lamda times open b right arrow dot operator c prime right arrow close

However, equation left hand side open a right arrow multiplication open b right arrow multiplication c right arrow close dot operator c prime right arrow close equals right hand side open c prime right arrow multiplication open b right arrow multiplication c right arrow close dot operator a right arrow close

equation sequence absolute value of b right arrow multiplication c right arrow equals absolute value of b right arrow times absolute value of c right arrow times sine of times times b right arrow c right arrow hat equals absolute value of b right arrow times absolute value of c right arrow times cosine of times times b right arrow c prime right arrow hat

MathJax failure: TeX parse error: Missing or unrecognized delimiter for \right therefore

sine of multiplication multiplication times times c prime right arrow b right arrow c right arrow hat equals one

and

equation sequence absolute value of c prime right arrow multiplication open b right arrow multiplication c right arrow close equals absolute value of c prime right arrow times absolute value of b right arrow times absolute value of c right arrow times cosine of times times b right arrow c prime right arrow hat equals absolute value of c right arrow times open b right arrow dot operator c prime right arrow close

Hence equation left hand side c prime right arrow multiplication open b right arrow multiplication c right arrow close equals right hand side c right arrow times open b right arrow dot operator c prime right arrow close

equation left hand side open a right arrow multiplication open b right arrow multiplication c right arrow close dot operator c prime right arrow close equals right hand side open b right arrow dot operator c prime right arrow close times open a right arrow dot operator c right arrow close

equation left hand side open b right arrow dot operator c prime right arrow close times open a right arrow dot operator c right arrow close equals right hand side lamda times open b right arrow dot operator c prime right arrow close

lamda equals open a right arrow dot operator c right arrow close

mu can be calculated in a similar manner, however, it is easier achieved using equation open asterisk operator close .

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