Projections

Anyone know where they get W.v = &v.v + u.v = &v.v on page 731?

Thanks

start with: w = &.v + u
you can get this from plain vector addition of the the parallel (&.v) and orthogonal (u) components of w. or the "triangle rule" if you prefer.

then multiply both sides by v, for no good reason other than to get the result: w.v = &.v.v + u.v

now the u.v = 0 part comes into play - you can substitute u.v in the previous equation with zero, giving you w.v = &.v.v as advertised. As it says in the textbook, u.v = 0 because u and v are defined as being orthogonal and the dot product of two orthogonal vectors is zero.

for the statement

"then multiply both sides by v, for no good reason other than to get the result: w.v = &.v.v + u.v"

Vector dot product is not the same as vector multiplication.

But Would that mean you take the dot product of both sides?

I guess so. I kindof equated the two statement like I would have in highschool: multiply both sides by the same thing and the equality holds. I don't know my math well enough to know if taking the dot product on both sides yields the same result, but it certainly looks that way.

http://tutorial.math.lamar.edu/AllBrowsers/2318/OrthonormalBasis.asp

is a good practise site

Any idea why on p271 they chose to use P=ST instead of P=TS, which is simpler? i.e. are they not equivalent?

also. on p248 do you think that when they say the plane formed by n and v_up that they mean the plane formed by the point vrp and these two vectors? and if so how is that different from the plane that we're trying to find a reference frame for? and if it is different why should v lie in that plane. and why should v be a linear combination of n and v_up because of this? and why can we set beta=1?

dont think we have to know this

I think we have to know the assignment maths , maths in the tutorials and the appendixes.