IIT JEE , AIEEE Forum - Mathematics , Vectors

jv604

1). if vector a, vector b ,vector c are the position vectors of the non- collinear points A,B,C resp. then show that a*b+b*c+c*a is perpendicular to the plane of ABC.(* is actually cross).,where a,b,c are vectors.

2). Let a ,b,c be 3 non zero vectors. if a.(b*c)=0 and b,c are not parallel then prove that a=

b+kc where

and k are some scalars.

3). if a,b,c represent the vectors BC,CA and AB of a triangle ABC then show that a*b=b*c=c*a.

(* is actually cross product in vectors in all 3 sums)

ayshwarya

ans 4 3rd 1 a=OC-OB , b=OA-OC , c=OB-OA hence by cross product v wud get OA*OB+OB*OC+OC*OA

nishantsingh89

1)
as a,b,c lie in same plane so
a*b is perp. to the plane ABC
b*c is perp. to the plane ABC,
similarly for c*a...

as a*b, b*c, c*a all lie in plane perpendicular to tht of ABC,
so a*b +b*c+c*a lies in plane perpendicular to tht of ABC

aliter

let a*b+b*c+c*a lie in any plane perp to ABC
then its dot product with any of a,b,c shud be zero
on taking dot product with a
we get [aba] + [bca] + [caa]
[aba]=[caa]=0
and since b,c,a are coplanar so [abc] =0
thus dot product is zero so the vector lies in plane perp to ABC

studyid

hey jv604,
see i have a sol. to yr 3rd q.
it is as follows :

see vector a =BC vctor, then vector b = AC vector and lastly vctor c=AB vctor.

now since these three vectors a,b n c for a

, it is very obvious acc, to

law of vector addition, that a+b+c=0.

Now thats a imp. relation.

now take cross product of a on both the sides,

a x (a+ b + c) = a x 0
a x b + a x c =0
axb=cxa.----(1)

now take cross product on both sides with b

b x (a+b+c)=b x 0
bxa+bxc=0
bxc=axb---(2)

hnce by 1 & 2,
axb=bxc=cxa

ayshwarya

4 2nd quesn lets cum from end its a=lambda b +kc hence

[ ( lb+kc)*b] . c

= k(c*b) . c =0 hence proved

nishantsingh89

2) a.(b*c)=[abc] =0

thus abc are coplanar vectors

and if three vectors lie in a plane then any vector can be expressed as a linear combination of other 2 vectors

puneet

hii

well u can ask only one question ... neways .. most of the answers are already there .. lemme tell something more about the 2nd question ... the vector (b*c) is not in the plane of b and c .. infact perpendicular to that plain ...

now, since the dot product of a with (b*c) is zero .. a must be in plain of b and c

and hence the answer

cheers

Ask & Discuss Questions with Community & Experts