Logic behind some maths

[K.I.L.E.R]

I'm trying to derive loads of different maths formula and stuff that I memorised, but I'm going back and deriving everything in order to better my understanding of modelling using math.

My problem however isn't mathematical as such, it's logical.

I've gone through vector -> vector projections and found an anomyly I cannot logically grasp.

The standard vector projection example goes along the lines of:
w = .7071, .7071
v = 1, 0
v.t = w.v

My working:
u = w + v.t
t = (w-u)/v

t.v = w.v
u = w - w.v
t = (w - (w-w.v)) / v
u = w - ((w-(w-w.v))/v)*v

My book shows the formula: u = w - (w.v.v)/v.v

My problems:
My final equation for 'u' has a weird look (w-(w-w.v), however when I tested this with numbers I got the exact same answer as (w.v.v) so I figure it's just the way I worked it out. Not really a problem, more of a 'why?'.

t = (w-u)/v

What I can't understand is why "t = (w-u)/v.v" and not "t = (w-u)/v".
Logically you can determine that because you are looking for a scalar you may not have any vectors or points in your equation, so logically you'd get rid of it by making a dot product of itself, however it makes little sense when you think of it logically because you can't just go around and start changing things without a reason.

v.vt = v.v is a very strange relationship.
I understand that v.vt = vt because v.w is a projection vt and vt.v is a projection onto itself, therefore nothing's changed.

Can someone please explain this strange behaviour?

 

[Basic]

You're mixing up vectors and scalars.
Try it one more time, but use bold for all vectors, a (bold) dot for dot product, and an asterisk for vector times scalar. Then you'll probably see for yourself what mathematical errors you've done.

I'll give you a start. Looking at the formula from your book, it seems as you want to project the vector w along the vector v onto the plane 0 = v . x

So your equations are:
u = w + v * t
0 = v . u

Solve u, which means that you have to solve t first.

... and I'll leave the rest to you.

Remember that you can't divide by a vector, and you can't have vector=scalar.

 

[KimB]

You can't divide by a vector.

 

[K.I.L.E.R]

So is that the reason you have to divide (instead of the vector) by the vector dot product of itself?
I believe that a vector dot product of itself is just a projection of itself onto itself, therefore v.v = v * v.v?

You can't divide by a vector.

I will definitely redo the problem as many times as it takes in order to get it right.
It's cruicial to understand rather than to memorise and use.
Thanks to you both.

 

[KimB]

So is that the reason you have to divide (instead of the vector) by the vector dot product of itself?
I believe that a vector dot product of itself is just a projection of itself onto itself, therefore v.v = v * v.v?

No, because the left hand side of that equation is a scalar, and the right is a vector.

 

[K.I.L.E.R]

Just a quick question, I've never thought of this until recently but what is the abstract meaning of an inner product?
Wikipedia and the Internet have been useless in giving me an answer to this and my engineering book claims it has importance in function analysis however I cannot find anything out about it, everything to do with inner products is immediately linked to pre-Hilbert spaces in geometry.

 

[Fred]

Yea mathematically an inner product, together with a vector space on which its defined, is called an inner product space. This space is normed in the obvious way, and allows you to define notions of length (eg it induces a metric).

In mathematics, the inner product need not be euclidean, and this is one of the key concepts to make the step to different geometries.

 

[Basic]

So is that the reason you have to divide (instead of the vector) by the vector dot product of itself?

Not just like that. Since you can't divide by a vector, you'll have to find another way. And in this case, this other way will include dividing with (v . v). Here's how it's done:

The two original equations:
u = w + v * t
0 = v . u

Insert the first one into the second:
0 = v . u = v . (w + v * t) = v . w + v . v * t
<=>
t = - (v . w) / (v . v)

And then insert this t into the first equation:
u = w + v * t = w - v * (v . w) / (v . v)

I believe that a vector dot product of itself is just a projection of itself onto itself, therefore v.v = v * v.v?

No, the dot product of a vector with itself is the square of the length of the vector.

It's cruicial to understand rather than to memorise and use.

True. A big part of why I am the person I am, and why my friends seem to think of me as a rather smart person (sorry for the bragging), is that I have a quite bad memory. So I have to compensate that by being good at deducing stuff. Like, I'm fairly good at maths, but I don't know how to do a partial integration . I can however deduce how to do it in less than 5 seconds. Knowing it in that way meant that when I attended a vector analysis course, I already "knew" a lot of vector-variants of partial integration without even knowing it .


And now some descriptions of various expressions:

v . v = square of length of v

Let e = v / sqrt(v . v) = v normalized, ie same direction but length 1

e . w = length of w projected onto e
e * (e . w) = project w onto e
v * (v . w) / (v . v) = project w onto v, division by (v . v) to compensate that v isn't normalized.

Projecting w onto the plane 0 = v . u is the same as w minus the projection of w onto the planes normal v.
u = w - v * (v . w) / (v . v)
Oooh, that's the same expression as before, but this time from reasoning and knowing what some expressions means geometrically instead of just doing pure math. (It can be good to know both ways.)