VCE Specialist 3/4 Question Thread!

[brightsky]

The dot product exists so that we can find the angle between two vectors. Recall the formula a.b =  |a||b| cosΘ.

Also, a bit nit-picky, but finding the dot product of two vectors is different from multiplying two vectors. You cannot multiply vectors. You can find various products of two vectors though (dot product, cross product, etc.).

[7equals7]

Complex Number Question!

Can someone help me with this proof?

using the result of



prove that

[lzxnl]

Complex Number Question!

Can someone help me with this proof?

using the result of



prove that

so you just have
and the result follows

I think you should have had conjugate of z then raised to the power of n as the thing to prove equivalence with

[lzxnl]

Just started 3&4 spec beginning with vectors and my only knowledge about them is from completing 3&4 physics so my understanding of concepts is very weak.
What is the purpose of the dot product of two vectors? I understand why to add vector but I don't understand the purpose of multiplying them.
Could someone explain in very basic terms, thanks

As for dotting vectors together
One application, as mentioned by brightsky, is to work out the angle between two vectors.
Another application is to multiply the scalar component of one vector in the direction of the other.
Draw up a diagram.
If you have vectors a and b, with t being the angle between a and b, then |a cos t| = magnitude of component of a parallel to b
The magnitude of the component of b parallel to b is, well, just b. Therefore the product of the components is |ab cos t|, also known as the dot product.
Why is this useful? I'll give an example, even though this is strictly speaking outside VCE requirements for anything.

In physics, we're taught that work equals force times distance, for a constant force. However, strictly speak work is a dot product as well. Only the component of the force parallel to the displacement actually speeds up or slows down the particle; in VCE, we also learn that uniform circular motion arises due to a force always perpendicular to the motion, and as it's uniform motion, the speed is constant. Thus, a force always perpendicular to the motion of an object won't change the object's speed or its kinetic energy, and thus does no work. Therefore, if you resolve the force into components perpendicular and parallel to the displacement, you'll see that only the parallel component is involved in the work. This means that to find out the work done, you actually take a dot product of the force and displacement vectors.

I could go on and on about where dot products are used in physics, but that would be long-winded, boring and not especially helpful. Just appreciate that the dot product has its uses and that it's needed for VCE (:

[Eugenet17]

Can someone help me find the asymptotes for this hyperbola

I'm getting a different answer and I don't know why :x

[abcdqd]

Can someone help me find the asymptotes for this hyperbola

I'm getting a different answer and I don't know why :x

The equations for the asymptotes are:
Here we have and and
So the asymptotes are

You can do the rest

[Eugenet17]

OH MY GOD I forgot about the 4 hahaha, thanks!

[lzxnl]

Don't remember formulas. Just...don't. It's not conducive to learning how these asymptotes are found.

OK, so we have 4/3 (x-1)^2 - (y-1)^2/3 = 1
(y-1)^2 = 4(x-1)^2 - 3
=(x-1)^2 * (4 - 3/(x-1)^2 )

As x and y both become large, the 3/(x-1)^2 term becomes arbitrarily close to zero. Thus we can see that (y-1)^2 approaches (x-1)^2 * 4
Find asymptotes from this

[zhe0001]

Please help by giving me a starting point or a hint

If , prove that

Thanks!

[kamil9876]

Have you found the real and imaginary parts of the expression? If so let and then apply the trig double angle formulas to the real and imaginary components.

[zhe0001]

Have you found the real and imaginary parts of the expression? If so let and then apply the trig double angle formulas to the real and imaginary components.

ok, but I still don't understand. Not sure what double angle formulas are as well. Searched it up, but not sure where to apply.

[SocialRhubarb]

Double angle formulae:





You can show that these two are the same using the identity

Hint:

[Kuroyuki]

So when sketching graphs like the one below
f(x) = (3x^2 + 2x - 26)/ (x^2 + x -6)
I don't understand why the vertical asymptote of y=3 is crossed on one side of the graph at (8,-3)
If its crossed it, what exactly is it then approaching?
Thanks

[mackintosh]

So when sketching graphs like the one below
f(x) = (3x^2 + 2x - 26)/ (x^2 + x -6)
I don't understand why the vertical asymptote of y=3 is crossed on one side of the graph at (8,-3)
If its crossed it, what exactly is it then approaching?
Thanks

? (8,-3) is not a point on f(x). y=3 is a horizontal asymptote that is never crossed

[devilsadvocate]

Can you please help me with this question? I've been stuck on it for ages.

"Prove that the diagonals of a parallelogram bisect each other." (It's a vector proof question.)

Thanks in advance!