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:
,
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 is perpendicular to the plane of and , must lie in this plane, therefore:
Dot-multiply both parts by :
Since , left-hand side is 0, so:
Now define vector lying in the plane of and , perpendicular to and directed so that , and form the left-hand oriented system. This guarantees that the angle between and , .
Dot-multiply both parts by :
However,
MathJax failure: TeX parse error: Missing or unrecognized delimiter for \right therefore
and
Hence
can be calculated in a similar manner, however, it is easier achieved using equation .
Vector horror
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:
,
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 is perpendicular to the plane of and , must lie in this plane, therefore:
Dot-multiply both parts by :
Since , left-hand side is 0, so:
Now define vector lying in the plane of and , perpendicular to and directed so that , and form the left-hand oriented system. This guarantees that the angle between and , .
Dot-multiply both parts by :
However,
MathJax failure: TeX parse error: Missing or unrecognized delimiter for \right therefore
and
Hence
can be calculated in a similar manner, however, it is easier achieved using equation .