Maths help

[LoveToStack]

I need all round help on this one, cannot do the first part of the question but I've made an attempt at the second part which I'm convinced is incorrect.

PQRS is a regular tetrahedron of side 1 unit.
PQ, PT, PR, and PS are all representatives of u, v, w, and x repsectively.
Point T is the Midpoint of RQ.

http://i46.tinypic.com/2q1cmr6.png

Express v in terms of u and w and show that: x.v = 1/2

I know which angles are 60 degrees and all that, it's just the first bit where you have to express v in terms of u&w that's getting me. I thought it might just be v= 1/2(u.w) since T is the midpoint of RQ. However if that was the case then x.v = 1 x 1/4 x cos(60) = 1/4 x 1/2 = 1/8 and it's meant to be 1/2.
It's driving me mad. Hate vectors. Any help greatly appreciated.

[N!ck]

v = 1/2 (u + w)

Can you do the dot product part from here?

[LoveToStack]

I see exactly what you mean, because you can make the connection about parallel vectors so u+w=2v. Plugging that in you'd get:
x.v= 1 x 1/2(u+w) x cos60 = 1 x 1/2(1+1) x cos60 = 1 x cos60 = 1/2
Works perfectly! :) Thankyou very much indeed. +Rep
Argh, need to spread, +rep owed.

[LoveToStack]

Right I need more help with vectors but it seems wasteful to start a new thread. The question is:

VABCD is a rectangular based pyramid.

AB = 8, 2, 2 (imagine these are components, not co-ords obviously)
AD = -2, 10, -2
AV = 1, 7, 7

Express CV in component form.

Click the image to enlarge.

22131

I don't know where to begin at all, just have a really poor understanding of vectors. Anyway I can work out AC using AC = AB + AD, which gives me 10, 12, 0. From that I know that midpoint AC is 5, 6, 0.
However that's where my progress stops, I have no idea where to go from there, or if that information is relevant.

Wait a sec, had a brainwave after posting. If we call the midpoint of AC, M, could I work out MV saying that MV = AV - AM? Then once I have MV, I could say that CV = MV - AM?
That might work but it's unlikely considering how poor I am at vectors. If anyone can see that working out, or knows a better way then please say, help greatly appreciated. Ill try the aforementioned way now.

Doing that method I got an answer, which is progress enough lol, but confirmation is still appreciated.

[N!ck]

You should be able to work out CV by following components of a path from C to V that you have the data for. It involves AB AD and AV that you have been given. Some can be negative. I'll put a big hint in a spoiler.

- AD - AB + AV

Because you know that AD = BC due to symmetry.

[LoveToStack]

Yeah I know that they're symmetrical so share the same components and I used that to get AC. Once I had AC I did a kind of roundabout way of getting to the answer, but expressing CV in terms of everything else I got:

CV = AV - 1/2AC

That fit? I have components as the Q requires btw, but the expression covers that.

EDIT: dear god, I see what you mean now. So simple. >:l
So it would just be -AD - AB + AV. Oh man that's so simple, I am such a fool. Cheers for your help again, much appreciated. Ignore the rest of this post lol.

[N!ck]

Your brain wave should actually be correct as well. Just a far more complicated way of thinking about it.

You have calculated AC incorrectly in your previous post. However, with that corrected your method should give the same answer. For some reason it didn't when I calculated it with the numbers but I was working in my head and probably missed something.

[LoveToStack]

Ah your right AC should have been 6, 12, 0. I went through it again just to see if it would have actually worked and it does work out ok but as you say it's way more effort than is necessary. Thanks again for all your help, it's been great and I feel a lot more comfortable with vectors now.

[N!ck]

Ah your right AC should have been 6, 12, 0. I went through it again just to see if it would have actually worked and it does work out ok but as you say it's way more effort than is necessary. Thanks again for all your help, it's been great and I feel a lot more comfortable with vectors now.

Be glad you're not studying vector calculus like I am at the moment. Now that causes confusion :P.

[LoveToStack]

I plan to take pure maths next year lol then maybe at university as well so I'll savour this stuff while it's available.