Vectors q. (1 Viewer)

[astj]

Answers say that B = -1/2 + 5/2i D = 9/2 + 1/2i but I don't know how to get there

Thanks in advance

 

[Tiajfe9drffdrf]

There might be a shorter way but this is how I do it

 

[astj]

ah ok thanks I forgot to scale it so my answer was off, thanks!

 

[Luukas.2]

Another approach is to say that AB = i.AD = DC. Thus, AC = AD + DC = AD + i.AD = (1+ i).AD.

Now, AC = AO + OC = OC - OA = 3 + 4i - (1 - i) = 2 + 5i.

With these two results, it follows that:

​

 

[ZakaryJayNicholls]

Answers say that B = -1/2 + 5/2i D = 9/2 + 1/2i but I don't know how to get there

Thanks in advance

There are actually heaps of ways to answer these kinds of questions.

This can definitely be done super quickly because it is a square.

Midpoint of AC is (2, 1.5i), slope of AC is 5/2, and vector from A to AC midpoint is (1, 2.5i).

This means if you add (2.5,-1i) to the midpoint of AC you will get D [4.5, 0.5i] and if you subtract it, you will get B [-0.5, 2.5i].

Note:

We should note that the diagram is unfortunately not to scale, which you should always assume anyway, but makes the answers feel wrong.

 

[Luukas.2]

There are actually heaps of ways to answer these kinds of questions.

This can definitely be done super quickly because it is a square.

Midpoint of AC is (2, 1.5i), slope of AC is 5/2, and vector from A to AC midpoint is (1, 2.5i).

This means if you add (2.5,-1i) to the midpoint of AC you will get D [4.5, 0.5i] and if you subtract it, you will get B [-0.5, 2.5i].

Note:

We should note that the diagram is unfortunately not to scale, which you should always assume anyway, but makes the answers feel wrong.

This approach effectively turns the question into a problem in coordinate geometry... if A(1, -1) and C(3, 4) are the ends of a diagonal of square ABCD find coordinates for B and D.

If asked as a MCQ, it is certainly a quick approach.

However, for a written question, and given the application of coordinate geometry methods to the Argand Diagram is not covered by the syllabus, there is a danger of a marker taking a view that results from coordinate geometry are being assumed without sufficient justification.

I agree with ZJN that there are many approaches that will work here, but advise thinking carefully about what needs to be included when taking approaches that are atypical or even wander outside the formal syllabus.

Another approach, incidentally, would be to let and be the required position vectors, and then form a pair of simultaneous equations:

​

 

[ZakaryJayNicholls]

"
However, for a written question, and given the application of coordinate geometry methods to the Argand Diagram is not covered by the syllabus, there is a danger of a marker taking a view that results from coordinate geometry are being assumed without sufficient justification.
"

This is only true if it is specified in the question that you are to use a specific method, it is not true in general.

The math curriculums do not work in an isolated fashion, coord geometry has already been learnt in 7-10 and some 11-12 units, it is assumed knowledge. Any known or obvious results stemming directly from prerequisite content are typically accepted without justification.

When the markers mark HSC question (e.g. the only questions which matter) they look for correctness of working, not strict adherence to the 11-12 syllabus. This is why the marker guidelines specify marks for specific components of an answer or equivalent merit. In fact, many of the best textbooks (e.g. Terry Lee, Aus & Fitzpatrics, Pender, Grove, Howard, etc) routinely advocate use of non-curriculum methods as preferential methods for solving specific equations (typically when some other method is easier - the reason being that this is how formal mathematics operates).

Using a more complicated method where an easier one would have worked is of no value in HSC mathematics, VET mathematics, or university mathematics (as someone who has taught all of these curriculums for over a decade, I am fairly familiar with which things earn marks and which things don't). In fact, the most superior students will not only know how to do each question, but they will also have identified the fastest and easiest way, this is because simple methods save time.

 

[liamkk112]

This is only true if it is specified in the question that you are to use a specific method, it is not true in general.

The math curriculums do not work in an isolated fashion, coord geometry has already been learnt in 7-10 and some 11-12 units, it is assumed knowledge. Any known or obvious results stemming directly from prerequisite content are typically accepted without justification.

When the markers mark HSC question (e.g. the only questions which matter) they look for correctness of working, not strict adherence to the 11-12 syllabus. This is why the marker guidelines specify marks for specific components of an answer or equivalent merit. In fact, many of the best textbooks (e.g. Terry Lee, Aus & Fitzpatrics, Pender, Grove, Howard, etc) routinely advocate use of non-curriculum methods as preferential methods for solving specific equations (typically when some other method is easier - the reason being that this is how formal mathematics operates).

Using a more complicated method where an easier one would have worked is of no value in HSC mathematics, VET mathematics, or university mathematics (as someone who has taught all of these curriculums for over a decade, I am fairly familiar with which things earn marks and which things don't). In fact, the most superior students will not only know how to do each question, but they will also have identified the fastest and easiest way, this is because simple methods save time.

i'll have to agree with luukas on this one, to an extent. teachers can certainly give you marks for out of syllabus methods, or methods not supplied in the marking criteria, hence the equivalent merit. however it is always better to stick within the syllabus as much as possible.

good example is l'hospitals rule. you can solve some limits in seconds, so its great that it is fast and easy; however you run the risk of a teacher not fully rewarding your work. this is because now your marking criteria is in the hands of the marker, and the criteria supplied likely has no similarlities to your method, so there are really no rules anymore. they are free to deduct a mark if they deem that there is insufficient work, so say that you do not write down the formula for l'hospital's rule and simply state that you have applied it, they may decide that this is not enough working and this may cause you to lose a mark - essentially you now have to tread much more carefully than if you played inside the syllabus.

however, in the case of the coordinate geometry you applied to the question, i would say it should be fine as it is technically prerequisite content, but the working out should definetly be much more detailed. this way, the marker couldnt argue any reason to deduct marks. this could include giving reasoning to why adding the values to the midpoint gives B and D, as unlike the vector method you now need to provide geometric reasoning as to why this is, since you are now working with coordinate geometry; otherwise, the marker could deduct marks for insufficient working.

 

[Luukas.2]

This is only true if it is specified in the question that you are to use a specific method, it is not true in general.

The math curriculums do not work in an isolated fashion, coord geometry has already been learnt in 7-10 and some 11-12 units, it is assumed knowledge. Any known or obvious results stemming directly from prerequisite content are typically accepted without justification.

When the markers mark HSC question (e.g. the only questions which matter) they look for correctness of working, not strict adherence to the 11-12 syllabus. This is why the marker guidelines specify marks for specific components of an answer or equivalent merit. In fact, many of the best textbooks (e.g. Terry Lee, Aus & Fitzpatrics, Pender, Grove, Howard, etc) routinely advocate use of non-curriculum methods as preferential methods for solving specific equations (typically when some other method is easier - the reason being that this is how formal mathematics operates).

Using a more complicated method where an easier one would have worked is of no value in HSC mathematics, VET mathematics, or university mathematics (as someone who has taught all of these curriculums for over a decade, I am fairly familiar with which things earn marks and which things don't). In fact, the most superior students will not only know how to do each question, but they will also have identified the fastest and easiest way, this is because simple methods save time.

The point that I was making was not whether the method being used works - as it clearly does - but whether a student's response might be judged as showing all necessary working, especially in an assessment task where individual marking can be considerably more variable than HSC marking.

The method that you are using is, in effect, taking a problem on the Argand Diagram, converting it to an equivalent problem on the Cartesian Plane, solving it, and then translating that solution back to the Argand Diagram. Supporting those steps sufficiently calls for judgement from the student and immediate comprehension of a potentially unanticipated / unfamiliar approach from the marker. Whether you accept this as true or not, it is a fact that there is potential for an inadequately supported answer or a misinterpretation from the marker. I recognise that your use of vectors is actually staying with the Argand Diagram, but that will not be the case for many using this style of approach. Your method also converts the vector (1, 2.5i) to (2.5, -i) without explanation, which could be argued to be skipping necessary working.

I also suggest that your assertion that "the most superior students will not only know how to do each question, but they will also have identified the fastest and easiest way, this is because simple methods save time" is an oversimplification. There is not one unique "simplest" solution for every problem. A question might be solvable by an algebraic or geometric method, and Student A may find algebraic approaches simpler whilst Student B finds geometric reasoning simpler. Telling a class with Students A and B that one method is faster / easier / simpler disregards individual differences and competencies, which is potentially a disservice to both students and others in the class. Applying a longer method correctly and confidently may actually be faster for some students. For example, recognising and apply reasoning like

​

is providing a solution more efficiently than if the full auxilliary angle method is used, but I would encourage a student to provide the full auxilliary angle method working and get to the correct answer if their attempts to use the short-cut does not reliably lead to the correct answer.

 

[ZakaryJayNicholls]

So in the 2023 Ext 1 HSC Q14ci it says "or otherwise"

View attachment 42119
So would you say the following gets 3 marks? It is 100% correct by the way:




The simplicity of this is in stark contrast to the rather lengthy and complicated solution found in https://educationstandards.nsw.edu....-d8a06d44-7551-4623-8cb7-e1cb2b472874-oLbc9Nm

That’s an interesting question, I believe cross products is included in extension2 and is never taught in extension1 unless as an optional flex topic. I often teach cross products to the ex1 students I tutor, as doing dot and cross together makes sense to me, and the university calc1/calc2 courses they will take will teach intro vectors including dot/cross/etc very quickly, so it’s better for them to have seen the actual math they will need as early as is possible.

The exclusion of cross products from extension 1 is a bit silly, as there are many cases involving areas where cross is simply vastly superior. I imagine the markers would not award full marks, as the method used is beyond the content of the ex1 course (never done prior, but to be done in future), but they should be giving full marks as the method not only works but also is done in an incredibly neat and concise way, which is what good math is all about.

 

[WeiWeiMan]

So in the 2023 Ext 1 HSC Q14ci it says "or otherwise"

View attachment 42119
So would you say the following gets 3 marks? It is 100% correct by the way:




The simplicity of this is in stark contrast to the rather lengthy and complicated solution found in https://educationstandards.nsw.edu....-d8a06d44-7551-4623-8cb7-e1cb2b472874-oLbc9Nm

I’m not entirely sure if that method gives full marks but I used a right angled triangle to find sin using cos and got 3/3 for it.

 

[Luukas.2]

That’s an interesting question, I believe cross products is included in extension2 and is never taught in extension1 unless as an optional flex topic. I often teach cross products to the ex1 students I tutor, as doing dot and cross together makes sense to me, and the university calc1/calc2 courses they will take will teach intro vectors including dot/cross/etc very quickly, so it’s better for them to have seen the actual math they will need as early as is possible.

The exclusion of cross products from extension 1 is a bit silly, as there are many cases involving areas where cross is simply vastly superior. I imagine the markers would not award full marks, as the method used is beyond the content of the ex1 course (never done prior, but to be done in future), but they should be giving full marks as the method not only works but also is done in an incredibly neat and concise way, which is what good math is all about.

The cross product is NOT part of the MX2 syllabus.

Its exclusion from both the MX1 and MX2 syllabi is a bit silly, I agree, but the fact is that it is not in either syllabus.

 

[s97127]

The cross product is NOT part of the MX2 syllabus.

Its exclusion from both the MX1 and MX2 syllabi is a bit silly, I agree, but the fact is that it is not in either syllabus.

Does it mean that I cannot use cross product in mx1 and mx2 solution? Will I get 0 mark if I use it?

 

[carrotsss]

Does it mean that I cannot use cross product in mx1 and mx2 solution? Will I get 0 mark if I use it?

iirc you’ll get full marks if you get the question right, but if you don’t you won’t get any marks

 

[WeiWeiMan]

iirc you’ll get full marks if you get the question right, but if you don’t you won’t get any marks

How about for a ‘show this’ question (like 14ci 2023)

 

[carrotsss]

How about for a ‘show this’ question (like 14ci 2023)

id say you’d probably get the marks, there really isn’t much point though imo like just play it safe and stick to hsc methods in the hsc

 

[s97127]

iirc you’ll get full marks if you get the question right, but if you don’t you won’t get any marks

Does this mean that I can apply anything outside the syllabus i.e Chebyshev inequality in my solution?

 

[Luukas.2]

Does it mean that I cannot use cross product in mx1 and mx2 solution? Will I get 0 mark if I use it?

It has the same implication as using anything that is not in the syllabus... if it is correct and valid, it should be credited, but no one can promise how an individual teacher may respond in marking an assessment task. You need to give careful consideration to what reasoning and evidence is needed to support your response so that "all necessary working" is provided.

 

[Luukas.2]

For example, in this case, the solution that @tywebb has posted would be much harder for an HSC marker to criticise if it stated what a cross-product is and included working showing the reasoning. Perhaps something like:

The cross product of any two vectors, , is itself a vector with direction perpendicular to the plane defined by the two vectors and magnitude equal to the area of the parallelogram with the two vectors as adjacent sides. In this case, we have:

​

Now, since the magnitude of this cross-product is the area of the parallelogram with OA and OB as sides, and as the diagonal AB bisects the parallelogram, this magnitude is also the double the area of the triangle OAB.

​

 

[liamkk112]

The point that I was making was not whether the method being used works - as it clearly does - but whether a student's response might be judged as showing all necessary working, especially in an assessment task where individual marking can be considerably more variable than HSC marking.

The method that you are using is, in effect, taking a problem on the Argand Diagram, converting it to an equivalent problem on the Cartesian Plane, solving it, and then translating that solution back to the Argand Diagram. Supporting those steps sufficiently calls for judgement from the student and immediate comprehension of a potentially unanticipated / unfamiliar approach from the marker. Whether you accept this as true or not, it is a fact that there is potential for an inadequately supported answer or a misinterpretation from the marker. I recognise that your use of vectors is actually staying with the Argand Diagram, but that will not be the case for many using this style of approach. Your method also converts the vector (1, 2.5i) to (2.5, -i) without explanation, which could be argued to be skipping necessary working.

I also suggest that your assertion that "the most superior students will not only know how to do each question, but they will also have identified the fastest and easiest way, this is because simple methods save time" is an oversimplification. There is not one unique "simplest" solution for every problem. A question might be solvable by an algebraic or geometric method, and Student A may find algebraic approaches simpler whilst Student B finds geometric reasoning simpler. Telling a class with Students A and B that one method is faster / easier / simpler disregards individual differences and competencies, which is potentially a disservice to both students and others in the class. Applying a longer method correctly and confidently may actually be faster for some students. For example, recognising and apply reasoning like

​

is providing a solution more efficiently than if the full auxilliary angle method is used, but I would encourage a student to provide the full auxilliary angle method working and get to the correct answer if their attempts to use the short-cut does not reliably lead to the correct answer.

summarises well what i have been trying to communicate