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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,

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and

Hence

can be calculated in a similar manner, however, it is easier achieved using equation .