VCE Specialist 3/4 Question Thread!

[TheAspiringDoc]

Welp, I'd do it by parts, but I think in future it'd be best to leave non-relevant maths questions out of the further/methods/specialist question threads.

Ok, as in start a new thread for the question? and what did you mean by parts?

[pi]

Integration by parts

And I've moved this into the spesh thread, definitely not a Further Maths level question.

[Conic]

You can write .

I did this because x-1 will be a factor of x³-1, as 1³-1=0 (Factor Theorem).

Using difference of cubes or long division, you get . So now



                     

                     

                     

[keltingmeith]

^That's really cool, ngl.

Thought it up, I've been learning integration recently.

I also forgot to mention this, but be careful when thinking up integration questions. Not all elementary functions have an antiderivative. For example,

[Chang Feng]

Thanks and another question:

For question 7.b) why do we do abs(vector V) and not just vector v being equal to 0
Thanks again.

[Chang Feng]

And I'm still sorta lost. How is velocity just speed with direction.
Thanks. Again. Much appreciated

[cosine]

If there's one thing with vectors I dont understand, it would be resolution of vectors.

Could someone please refine the theory behind this, I know how to apply it, but what is the point if I don't know why i'm doing it. I would appreciate any responses, thanks!

[cosine]

And I'm still sorta lost. How is velocity just speed with direction.
Thanks. Again. Much appreciated

Velocity cannot be stated as a scalar by itself, that is, 20m is not velocity. But, if you introduce the direction/time alongside with the speed, you are stating a velocity. 20m/s (20 meters per second) is telling us that every second we are moving 20m towards our goal, were as if I only said 20m, it could take me 1 year to reach our goal, but we wouldnt know because the direction isn't provided!

Hope it helped

[keltingmeith]

If there's one thing with vectors I dont understand, it would be resolution of vectors.

Could someone please refine the theory behind this, I know how to apply it, but what is the point if I don't know why i'm doing it. I would appreciate any responses, thanks!

You'll learn about this more in UMEP, so if you're struggling now, it might be worth waiting until then to see if that helps.

For now, think of it like this:

Any vector can be broken into a sum of different components. For example, for the vector u=3i+j, we can break it up into the two components 3i and j, with the amount of components depending on the dimension the vector sits in (a vector in 2D has two components, a vector in 3D has three components, etc.).

However, what if we wanted to change the axes? Say, we rotated the axes 45 degrees, so that the x-axis now lies along the unit vector ? Well, all of a sudden our x-component is off - but, we can find the new x-component by projecting u onto it.

So, we do our projection, and get that the x-component is now .

Now, as before, x+y=u, so this should still hold for our new axes. So, we get y=u-x=3i+j-2i-2j=i-j.

So, now we've rotated our axes, and found the new axes unit vectors that make up u. Why would we want to do this, though? Well, consider the curve with equation x^2-xy+y^2=1, shown below:

http://www.wolframalpha.com/input/?i=plot+x%5E2+-+xy+%2B+y%5E2%3D1

Normally, this is hard to sketch - sure, we know what an ellipse is, but that equation has never been seen before - but, if we define the curve as a set of vectors, we can use the above method to rotate the axes enough that the equation is something we know how to draw. Then, we just gotta rotate it backwards, and all of a sudden we can draw this curve with an equation we've never seen before!

Hope you understand some of that!

Velocity cannot be stated as a scalar by itself, that is, 20m is not velocity. But, if you introduce the direction/time alongside with the speed, you are stating a velocity. 20m/s (20 meters per second) is telling us that every second we are moving 20m towards our goal, were as if I only said 20m, it could take me 1 year to reach our goal, but we wouldnt know because the direction isn't provided!

Hope it helped

Sort of... You're right that velocity is a vector quantity, and so needs a direction. But, the "per second" doesn't define direction. Here's the way I think of it:
            With direction (eg, north)             Without direction
m                   Displacement                           Distance
m/s                  Velocity                                   Speed
m/s/s    Acceleration/deceleration                       N/A

The last one depends on if slowing down or not, and really deceleration can just be thought of as acceleration in the opposing direction to motion. Often, acceleration is just defined as the positive direction if no direction is specified.

The direction can be defined in a few ways - one way is as a vector (for example, a displacement vector of 3i+j), another is to just say "moved left".

[cosine]

You'll learn about this more in UMEP, so if you're struggling now, it might be worth waiting until then to see if that helps.

For now, think of it like this:

Any vector can be broken into a sum of different components. For example, for the vector u=3i+j, we can break it up into the two components 3i and j, with the amount of components depending on the dimension the vector sits in (a vector in 2D has two components, a vector in 3D has three components, etc.).

However, what if we wanted to change the axes? Say, we rotated the axes 45 degrees, so that the x-axis now lies along the unit vector ? Well, all of a sudden our x-component is off - but, we can find the new x-component by projecting u onto it.

So, we do our projection, and get that the x-component is now .

Now, as before, x+y=u, so this should still hold for our new axes. So, we get y=u-x=3i+j-2i-2j=i-j.

So, now we've rotated our axes, and found the new axes unit vectors that make up u. Why would we want to do this, though? Well, consider the curve with equation x^2-xy+y^2=1, shown below:

http://www.wolframalpha.com/input/?i=plot+x%5E2+-+xy+%2B+y%5E2%3D1

Normally, this is hard to sketch - sure, we know what an ellipse is, but that equation has never been seen before - but, if we define the curve as a set of vectors, we can use the above method to rotate the axes enough that the equation is something we know how to draw. Then, we just gotta rotate it backwards, and all of a sudden we can draw this curve with an equation we've never seen before!

Hope you understand some of that!

Sort of... You're right that velocity is a vector quantity, and so needs a direction. But, the "per second" doesn't define direction. Here's the way I think of it:
            With direction (eg, north)             Without direction
m                   Displacement                           Distance
m/s                  Velocity                                   Speed
m/s/s    Acceleration/deceleration                       N/A

The last one depends on if slowing down or not, and really deceleration can just be thought of as acceleration in the opposing direction to motion. Often, acceleration is just defined as the positive direction if no direction is specified.

The direction can be defined in a few ways - one way is as a vector (for example, a displacement vector of 3i+j), another is to just say "moved left".

Got lost right at the 45 degrees part, I sort of understand it, but also don't... I guess it'll only take some time!

Thanks anyways bro

[keltingmeith]

Got lost right at the 45 degrees part, I sort of understand it, but also don't... I guess it'll only take some time!

Thanks anyways bro

Something that might work easier:



Using the left side and the bottom, you can define the vector as moving across some i and some j. But, if you tilt your head so that that the vector v is now along the bottom (like your x-axis), it now moves along a different set of i and j.

[cosine]

Something that might work easier:

(Image removed from quote.)

Using the left side and the bottom, you can define the vector as moving across some i and some j. But, if you tilt your head so that that the vector v is now along the bottom (like your x-axis), it now moves along a different set of i and j.

Oh okay! So we are finding the vector u in the direction of vector v right?

Also, another question:

OABC is a parallelogram where OA = 3i + 2j and OC = 2i - 3j

Because we have O as one of our vertices, does that mean O is the origin, or is it just some random point in space?

[lzxnl]

Oh okay! So we are finding the vector u in the direction of vector v right?

Also, another question:

OABC is a parallelogram where OA = 3i + 2j and OC = 2i - 3j

Because we have O as one of our vertices, does that mean O is the origin, or is it just some random point in space?

Think of it as the origin if you like, but it's really just a random point

[cosine]

Think of it as the origin if you like, but it's really just a random point

Yeah I realised it worked both ways, thank you!

[Chang Feng]

Just wondering are we required to know about similarity and congruency for triangles in specialist maths???