Given that ` vec adot vec b= vec adot vec c , vec axx vec b= vec axx vec ca n d vec a` is not a zero vector. Show that ` vec b= vec cdot`

Show that ( vec a- vec b)xx( vec a+ vec b)=2( vec axx vec b)dot

If vec a , vec b , vec c are three given non-coplanar vectors and any arbitrary vector vec r in space, where Delta1=| vec rdot vec a vec bdot vec a vec cdot vec a vec rdot vec b vec bdot vec b vec cdot vec b vec rdot vec c vec bdot vec c vec cdot vec c| , Delta2=| vec adot vec a vec rdot vec a vec cdot vec a vec adot vec b vec rdot vec b vec cdot vec b vec adot vec c vec rdot vec c vec cdot vec c| Delta3=| vec adot vec a vec bdot vec a vec rdot vec a vec adot vec b vec bdot vec b vec rdot vec b vec adot vec c vec bdot vec c vec rdot vec c| , Delta =| vec adot vec a vec bdot vec a vec cdot vec a vec adot vec b vec bdot vec b vec cdot vec b vec adot vec c vec bdot vec c vec cdot vec c| , then prove that vec r=(Delta1)/ Deltavec a+(Delta2)/Delta vec b+(Delta3)/Delta vec c .

If vec a= hat i+ hat j+ hat k , vec b= hat i- hat j+ hat k , vec c= hat i+2 hat j- hat k , then find the vaue of | vec adot vec a vec adot vec b vec adot vec c vec bdot vec a vec bdot vec a vec bdot vec a vec cdot vec a vec cdot vec a vec cdot vec a| .

Prove that [ vec a, vec b, vec c + vec d] = [ vec a, vec b, vec c] + [ vec a, vec b , vec d] .

If vectors vec a , vec b ,a n d vec c are coplanar, show that | vec a vec b vec c vec adot vec a vec adot vec b vec adot vec c vec bdot vec a vec bdot vec b vec bdot vec c|=0

If vec a , vec b ,a n d vec c are three vectors such that vec axx vec b= vec c , vec bxx vec c= vec a , vec cxx vec a= vec b , then prove that | vec a|=| vec b|=| vec c|dot

If vec a , vec b , vec ca n d vec d are distinct vectors such that vec axx vec c= vec bxx vec da n d vec axx vec b= vec cxx vec d , prove that ( vec a- vec d)dot (vec b- vec c)!=0,

If the vectors vec a , vec b ,a n d vec c form the sides B C ,C Aa n dA B , respectively, of triangle A B C ,t h e n (a) vec a . vec b+ vec b . vec c+ vec c . vec a=0 (b) vec axx vec b= vec bxx vec c= vec cxx vec a (c) vec adot vec b= vec bdot vec c= vec c dot vec a (d) vec axx vec b+ vec bxx vec c+ vec cxx vec a=0

If vec a , vec b , vec c are unit vectors such that vec adot vec b=0= vec adot vec c and the angle between vec ba n d vec c is pi//3 , then the value of | vec axx vec b- vec axx vec c| is 1//2 b. 1 c. 2 d. none of these

Show that the vectors 2 vec a- vec b+3 vec c , vec a+ vec b-2 vec ca n d vec a+ vec b-3 vec c are non-coplanar vectors (where vec a , vec b , vec c are non-coplanar vectors)