matrix algebra help plox? (1 Viewer)

[hayabusaboston]

2.) Consider u=(1,1,1) and v=(1,0,-1)
a) compute dot product u.v and vectror w=uxv
b) compute volume of parallelpiped defined by u,w,v
c) consider volume of parallelpiped defined by w-u,u,v. Using properties of the dot and cross products, show that volume is equal to that of the parallelpiped from previous question. Is there geometric reason you expect this to be true?


HALP PLOX

dot product is 0, what is the vector w, just (0,0,0)? how do u then do b and c?

o shit hang on cross product different to dot.

i j k
1 1 0.. etc

so is w (-1,-2,-1)?

is volume in b) 2?
also no idea what it means w-u,u,v.... How does that work?

Is w-u,u,v like

w-(u/u/v) in matrix form? Haaasmmasdasdkjajsd idk what im doing atm lol trying to brain this

 

[He-Mann]

Feeling sympathetic...
_________________

I got w = (-1, 2, -1).

If you understand what b) meant, then you should understand what c) meant.

The volume of a parallelpiped defined by u, w, v is V = u ⋅ (w x v).

The volume of a parallelpiped defined by w-u, u, v is V' = (w-u) ⋅ (u x v).

We wish to show that these two volumes are equal. Using cross and dot product properties (which will be omitted),

V' = (w - u) ⋅ (u x v)
= w ⋅ (u x v) - u ⋅ (u x v)
= w ⋅ (u x v) - v ⋅ (u x u)
= u ⋅ (v x w) - 0.

Comparing the expression of V and V', we see that w and v are swapped which means V and V' are opposite in sign. So you take the absolute for the negative one which shows that their volumes are the same.

I'm not sure about the geometric interpretation. Try to draw out a rough diagram of u, w, v and w-u, u, v and compare volumes.

 

[hayabusaboston]

Feeling sympathetic...
_________________

I got w = (-1, 2, -1).

If you understand what b) meant, then you should understand what c) meant.

The volume of a parallelpiped defined by u, w, v is V = u ⋅ (w x v).

The volume of a parallelpiped defined by w-u, u, v is V' = (w-u) ⋅ (u x v).

We wish to show that these two volumes are equal. Using cross and dot product properties (which will be omitted),

V' = (w - u) ⋅ (u x v)
= w ⋅ (u x v) - u ⋅ (u x v)
= w ⋅ (u x v) - v ⋅ (u x u)
= u ⋅ (v x w) - 0.

Comparing the expression of V and V', we see that w and v are swapped which means V and V' are opposite in sign. So you take the absolute for the negative one which shows that their volumes are the same.

I'm not sure about the geometric interpretation. Try to draw out a rough diagram of u, w, v and w-u, u, v and compare volumes.

OH WTF LOL

I totally missed that, (w-u),u,v omg

Thanks for your help You know my other q's too? I got them, just wanted to know if I was right.

http://imgur.com/a/stANL


3.a) Answer r=(1,1,1)+s(1,0,1)+t(0,2,2), got there as I explained before. Not sure what u were adding with the !
b) I converted a) to cartersian and got x+2y-z=2, then found it does NOT contain (0,0,0) or (0,0,2)
c) To find the line L I said since it goes through (1,1,1) and is of the form r=r0+t, then line is r=(1,1,1)+(1,2,-1)t, and the (1,2,-1) is the normal to the plane from b) and a), using n=uv and giving u=(0,1,0) and v=(-1,2,1) from a)
d) I got sqrt(3)/3 but I followed random thing online which I now forgot what it was LOL

 

[He-Mann]

named the points (1,1,1), (1,0,1) and (0,2,2) as PQR, did Q minus P and R minus P to find vectors u and v, denoted P(1,1,1) as ro
then from

r=r0+su+tv I plugged in results.

P = (1,1,1), Q = (1,0,1), R = (0,2,2)

u = Q - P = (0,-1,0)

v = R - P = (-1,1,1)

Then,

r = r0 + su + tv = r0 + s(0,-1,0) + t(-1,1,1)

which is not

r = (1,1,1)+s(1,0,1)+t(0,2,2)?????

_____

Anyway, I'm not doing the computation. If you did similar to my explanation to b), then it's right. Reasoning of c) is good.

You can exercise your own judgement instead of asking others for confirmation. I know that you know that help may not be abundant in the future as it is now but thinking independently is worth it in the long run; just a reminder. I know you're smart enough (post history analysis) to know all the advice I've been trying to give you but your actions imply otherwise...

 

[hayabusaboston]

P = (1,1,1), Q = (1,0,1), R = (0,2,2)

u = Q - P = (0,-1,0)

v = R - P = (-1,1,1)

Then,

r = r0 + su + tv = r0 + s(0,-1,0) + t(-1,1,1)

which is not

r = (1,1,1)+s(1,0,1)+t(0,2,2)?????

_____

Anyway, I'm not doing the computation. If you did similar to my explanation to b), then it's right. Reasoning of c) is good.

You can exercise your own judgement instead of asking others for confirmation. I know that you know that help may not be abundant in the future as it is now but thinking independently is worth it in the long run; just a reminder. I know you're smart enough (post history analysis) to know all the advice I've been trying to give you but your actions imply otherwise...

Yeaaa I just dont feel like its constructive for me personally to so deeply engross oneself in maths a good 9 weeks out from exams. I like maths but do it as part of my career goal of maths teacher, I'm not one of those people who does maths for the fun of it haha.

Thats a difficult thing to convey. I enjoy maths and know I will definitely enjoy a maths career, its just when no exams are on I have MANY other hobbies I prefer doing rather than sitting and doing maths.

 

[hayabusaboston]

P = (1,1,1), Q = (1,0,1), R = (0,2,2)

u = Q - P = (0,-1,0)

v = R - P = (-1,1,1)

Then,

r = r0 + su + tv = r0 + s(0,-1,0) + t(-1,1,1)

which is not

r = (1,1,1)+s(1,0,1)+t(0,2,2)?????

_____

Anyway, I'm not doing the computation. If you did similar to my explanation to b), then it's right. Reasoning of c) is good.

You can exercise your own judgement instead of asking others for confirmation. I know that you know that help may not be abundant in the future as it is now but thinking independently is worth it in the long run; just a reminder. I know you're smart enough (post history analysis) to know all the advice I've been trying to give you but your actions imply otherwise...

sorry I made a mistake, I put P as (1,0,1), Q as (1,1,1), R as (0,2,2)

 

[He-Mann]

Yeaaa I just dont feel like its constructive for me personally to so deeply engross oneself in maths a good 9 weeks out from exams. I like maths but do it as part of my career goal of maths teacher, I'm not one of those people who does maths for the fun of it haha.

Thats a difficult thing to convey. I enjoy maths and know I will definitely enjoy a maths career, its just when no exams are on I have MANY other hobbies I prefer doing rather than sitting and doing maths.

Fair enough.

sorry I made a mistake, I put P as (1,0,1), Q as (1,1,1), R as (0,2,2)

Then you must have concluded that r = P + sQ + tQ and NOT r' = P+ su + tv because r has Cartesian equation x+2y-z = 2. However P(1,0,1) does satisfy it.

The point is that you did the right working out for a) saying that r = r0 + su + tv but you wrote down r = r0 + sv + tw where v and w are different to u and v, respectively.

 

[hayabusaboston]

Yeaaa all g got it thnx <3

 

[He-Mann]

Yeaaa all g got it thnx <3

No worries. Feel free to shotgun me in the mouth if I had taught you something wrong.