COPLANAR IS MEANT TO BE TAKEN CARE OFF!!!!!!!!!
let a,b,c,d represent coplanar vectors and sin(A)a + 2sin(2B)b + 3sin(3C)c  4d = 0 then find the least value of sin(A)^2 +sin(2B)^2 + sin(3C)^2 is....
1. 7/8
2. 8/7
3. 1
4. 2/3

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29 Answers
i am not getting it as a value
but as 16d^{2}/(a^{2}+4b^{2}+9c^{2})
and as metal mentioned if there is no restriction choose d=0 min=0
yes rohans ryt .....
my ques is complete ,,, this is ol dat is given
well in my first answer i included that ..
but 0 can be the minimum
just take a=b=c=i^ and d=0
they are coplanar .... :)
@DEEP U R AN OVERSMART GUY
THRE IS NO OPTION 0 AVAILABLE AND UR QUES WAS NOT CLEAR
MR CHALLANGER
no rohan u r not on the ryt track
wat i was able to do was....
Using coplanarity theorem of vectors....
sin(A) + 2sin(2B) + 3sin(3C) = 4
nao wat.....?
Hey deepansh ..
a,b,c =i^ are coplanar!!!!
and further there is no relation between a,b,c, and d!!!
we can indeed have a minimum value
u guys r not readin the ques properly
ye bhi to dekho ki minimum value nikalni kiski hai ..\
mr . rohan wat u ve found out is the minimum value of sin(A) + 2sin(2B) + 3sin(3C)
thats not mah ques...
cant sinA sin2B sin3C be all zero???
when d=0 and a,b,c=i^ we get
sinA + 2sin2B + 3sin3C =0
also observe that the quantity given to be found out >=0
here sin2a sin2b sin3c if all zero agrees with the above eqn and also gives the minimum possible value of this =0 ...!!
and further the answer i got is by application of coplanarity and a little bit of cauchy schwarz law ...
i am talking about this answer
16d^{2}/(a^{2}+4b^{2}+9c^{2})
yaar d is a vector representing a point and using coplanarity theorem
and finally i dint mak the ques ....
FIITJEE guys gav it in one of the AITS last year....
Guys,i help u in this question.myself faculty of maths,bansal classes..firstly by condition of coplanarity u have to get sum of coefficients is equal to zero.so value of sinA+2sin2B+3sin3C=4,after that u can assume two vectors in form of i^+2j^+3k^and sinA i^+sin2B j^+sin3C k^and now apply cauchey swartz inequality in it.u will find result directly..answer is 8/7..thankx.
if no 1 s gettin it then i hv a soln .
which i m unable to understand it....
if u guys want dt then i can do that.....
@deepanshu
A,B,C,D r the angles between what???
make the question more clearer
The question is not clear. Are Sin(A), Sin(B), Sin(C) independent of vectors a,b,c and d?
If that's so, here's my answer:
Let me choose the vectors such that a+2b+3c4d=0
Then Sin(A) = Sin(B)= Sin(C)=1, and maximum value=3.