VCE Specialist 3/4 Question Thread!

[HERculina]

Isn't monkeywantsabananas way the same as wat truetears was talking about? and thanks paulsterio, made me look at it a different way

[HERculina]

Actually i have another question (part b of the previous) : Hence, show that |z + w| is less or equal to |z| + |w| . This is called the triangle equality. 

[monkeywantsabanana]

Isn't monkeywantsabananas way the same as wat truetears was talking about? and thanks paulsterio, made me look at it a different way

Yeah it's the one TT had... Paul did it another way which I thought was more complex than what TT had so I decided to write it out so you can compare the different ways and choose for yourself.

[paulsterio]

Yeah, I didn't see it the way you saw it, so when I worked it through using TT's method I ended up with A LOT of a, b, c and d's because I expanded the |z+w|^2 side, which I probably shouldn't have done

But yeah, TT's method is the faster method in this case, sorry TT, you're too good

[TrueTears]

Actually i have another question (part b of the previous) : Hence, show that |z + w| is less or equal to |z| + |w| . This is called the triangle equality. 

ahh the famous triangle inequality, there are so many different ways to prove it, here's one of my faves that i tend to use (uses the inner product and the famous CS inequality)

and hence the result.

[paulsterio]

Lol, is there a way of doing it that doesn't involve using Vector Spaces?

[TrueTears]

yeah ofcoz heaps lol there's a cool geometric proof too

[HERculina]

Lol i don't really understand what u just did can you do a way that extends from part a of the question

[Mr. Study]

Decide whether this is linear dependent/independent.

.

Thanks!

[nubs]

A group of vectors are always linear dependent if la + mb = c if my memory is correct (it's been a while)
(there is an exception to this I think, maybe you need to ensure that they're not parallel - someone else can confirm this, but your textbook should tell you anyway)
l and m are just random constants btw

so yeah you will get:

4li + lj + 3k + 2mi - mj + 3mk = -4i + 2j + 6k

Then you equate all the vectors in the direction i, j and k

so for the ones with direction i:
4l + 2m = -4
2l + m = -4
j:
l - m = 2
k:
3l + 3m = 6
l + m = 2

So there we have 3 equations

Try and solve for l and m by choosing two of the equations

Say you picked the first two equations, once you get the solutions for l and m, sub them into the third equation to see if it actually equals 6

from observation, it looks like we won't find solutions that will satisfy all 3 of the equations simultaneously, so therefore, they are independent. If we did find solutions that satisfied all 3 of them, then the set of vectors would be linearly dependent

as you can tell, this is quite rushed, so I may have made some silly little errors or a massive one somewhere

[paulsterio]

From observation, they are independent. You can work it through, but once you have experience, you can look at the sizes of the numbers as well as their signs and have a feel for whether it's dependent or independent.

[trinh]

If I'm right, another way is to take those three vectors and put them into a 3x3 vector and find its determinant (using CAS lol).

If the determinant=0, then the set of vectors is linearly dependent; vice versa..

So using the vectors you've given, you can either enter each vector columnwise or rowwise; you'll get the same number...

With the vectors you've given, the determinant=-72

Therefore, the set of vectors are linearly independent.

[paulsterio]

trinh, i wish i knew that! D:

damn it!! i should have thought of that! it's genius!

[Hutchoo]

If I'm right, another way is to take those three vectors and put them into a 3x3 vector and find its determinant (using CAS lol).

If the determinant=0, then the set of vectors is linearly dependent; vice versa..

So using the vectors you've given, you can either enter each vector columnwise or rowwise; you'll get the same number...

With the vectors you've given, the determinant=-72

Therefore, the set of vectors are linearly independent.

Wow, so simple >.> Never thought of that LOL.

//feels like dumbass.

[Truck]

Hi friends, I'm having a bit of trouble with this one application of de moivre's theorem...

Could someone please show me their method to solve z^3 + i = 0.

Thank you!