A vector question

[chid]

Hi everyone.

Vector a=xi+yj+zk is perpendicular to vectors -i-j+5k and i-j+k, and has a magnitude of √14 units. Find a.

I know that using the dot product and magnitude x,y,z can be determined after solving simultaneous equations. I was just curious: can the cross product (not part of Spec course) be used in this case? (And if so, is it actually easier?)

Thanks. 

[enwiabe]

yes it can!

The cross product of -i-j+5k and i-j+k is a vector perpendicular to both vectors

                       |i    j  k|
Cross product =  |-1-1 5|  = i(-1 + 5) - j(-1 - 5) + k(1 + 1) = 4i + 6j + 2k = b (the new vector)
                       |1 -1 1|
Now you want a magnitude of sqrt(14) units. The magnitude of that vector is sqrt(4^2 + 6^2 + 2^2) = sqrt(16 + 36 + 4) = sqrt(56) = 2sqrt(14), so that magnitude is TWO multiplied by sqrt(14). Therefore you want to halve b to get the required vector = 2i + 3j + k. And we know that we got the right answer because it fell out so nicely. Rule #1 of VCE maths.
                       

[dcc]

The cross product will not always give ALL of the answers to this particular type of question, but you can generally solve these questions by solving the simultaneous equations:







Then multiplying the resultant unit vector(s) by to get the required vector(s).

[enwiabe]

Uh, actually, it will give you a scalar multiple of the answer to this question. Always.

[Neobeo]

Don't forget that there are two answers.

[midas_touch]

Do they even teach cross products as part of the course?

[Collin Li]

Nope, and it means that you can't use it on a Specialist paper (unless you lay the groundwork for it by first principles - good luck doing that ).

[enwiabe]

I reckon it'd be about as tedious as those 3 annoyingly stupid simultaneous equations.

[Ahmad]

I emailed VCAA regarding whether use of maths beyond the specialist course was allowed. I didn't get a reply. Does anyone know for certain whether it's allowed? I'm curious.

[Ninox]

we asked that once, since a good half of the MUEP cohort was in my class.
my spesh teacher told us a lot of things about what Phillip swedosh, the spesh chief examiner said regarding special techniques. Yes you can use techniques beyond the spesh course (integration by parts and vector cross product for example) provided you show all working and get EVERYTHING 100% correct. So if your answer and working are totally right, you're doing fine.
However, if your answer is wrong, you won't get many, if any, working out marks for using 'exotic' maths.
 

[unknown id]

would a = -2i-3j-k be an answer as well?

[enwiabe]

Yes 'twould. That is the vector going in the opposite direction at 180 degrees to 2i + 3j + k. But it is still a vector perpendicular to both the given ones and has a magnitude of sqrt(14).

[bucket]

i wish i had chosen spec =[
looks so fun.

[enwiabe]

Ah well, you can do it at uni!