For any three vectors ` vec a , vec b , vec c ,` prove that `| vec a+ vec b+ vec c|^2=| vec a|^2+| vec b|^2+| vec c|^2+2( vec adot vec b+ vec bdot vec c+ 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 vec a ,\ vec b ,\ vec c are unit vectors such that vec a+ vec b+ vec c= vec0 find the value of vec adot vec b+ vec bdot vec c+ vec cdot vec adot'

For any two vectors vec a and vec b , prove that: | vec a+ vec b|^2=| vec a|^2+| vec b|^2+2 vec adot vec b , | vec a- vec b|^2=| vec a|^2+| vec b|^2-2 vec adot vec b , | vec a+ vec b|^2+| vec a- vec b|^2=2(| vec a|^2+| vec b|^2) and | vec a+ vec b|^2=| vec a- vec b|^2 iff vec a_|_ vec bdot Interpret the result geometrically.

For any three vectors a ,\ b ,\ c prove that [ vec a+ vec b\ vec b+ vec c\ vec c+ vec a]=2\ [ vec a\ vec b\ vec c]dot

If vec a , vec b , vec c are unit vector, prove that | vec a- vec b|^2+| vec b- vec c|^2+| vec c- vec a|^2lt=9.

For any three vectors veca , vec b , vec c show that vecaxx( vec b+ vec c)+ vec bxx( vec c+ vec a)+ vec cxx( vec a+ vec b)= vec0

If vec a , vec b ,a n d vec c are unit vectors such that vec a+ vec b+ vec c=0, then find the value of vec adot vec b+ vec bdot vec c+ vec cdot vec adot

If vec a+vec b+vec c=0 , prove that (vec a xx vec b)=(vec b xx vec c)=(vec c xx vec a)

Three vectors vec a ,\ vec b ,\ vec c satisfy the condition vec a+ vec b+ vec c= vec0 . Evaluate the quantity mu= vec adot vec b+ vec bdot vec c+ vec cdot vec a ,\ if\ | vec a|=1,\ | vec b|=4\ a n d\ | vec c|=2.

If vec a , vec b , vec c are three non-coplanar vectors, prove that [ vec a+ vec b+ vec c vec a+ vec b vec a+ vec c]=-[ vec a vec b vec c]