Prove that `[[ vec a+ vec b, vec b+ vec c, vec c+ vec a]]=2[ [vec a, vec b, vec c]]dot`

Let vec a , vec b ,and vec c be any three vectors, then prove that [ vec axx vec b vec bxx vec c vec cxx vec a ]= [vec a vec b vec c]^2

Let vec r be a non-zero vector satisfying vec r dot vec a= vec rdot vec b= vec rdot vec c=0 for given non-zero vectors vec a , vec b and vec c dot Statement 1: [ vec a- vec b vec b- vec c vec c- vec a]=0 Statement 2: [ vec a vec b vec c]=0

Prove that [ vec l vec m vec n][ vec a vec b vec c]=| (vec l. vec a, vec l. vec b, vec l.vec c), (vec m. vec a, vec m. vec b, vec m.vec c),( vec n. vec a, vec n. vec b, vec n.vec c)| .

[( vec axx vec b)xx( vec bxx vec c)( vec bxx vec c)xx( vec cxx vec a)( vec cxx vec a)xx( vec axx vec b)] is equal to (where vec a , vec ba n d vec c are nonzero non-coplanar vector) a. [ vec a vec b vec c]^2 b. [ vec a vec b vec c]^3 c. [ vec a vec b vec c]^4 d. [ vec a vec b vec c]

Let vec a , vec ba n d vec c be three non-coplanar vectors and vec p , vec qa n d vec r the vectors defined by the relation vec p=( vec bxx vec c)/([ vec a vec b vec c]), vec q=( vec cxx vec a)/([ vec a vec b vec c])a n d vec r=( vec axx vec b)/([ vec a vec b vec c])dot Then the value of the expression ( vec a+ vec b)dot vec p+( vec b+ vec c)dot vec q+( vec c+ vec a)dot vec r is a. 0 b. 1 c. 2 d. 3

Let vec a , vec b ,a n d vec c be vectors forming right-hand traid. Let vec p=( vec bxx vec c)/([ vec a vec b vec c]), vec q=( vec cxx vec a)/([ vec a vec b vec c]),a n d vec r=( vec axx vec b)/([ vec a vec b vec c]),dot If x in R^+, then a. x[ vec a vec b vec c]+([ vec p vec q vec r])/x has least value =2 . b. x^4[ vec a vec b vec c]^2+([ vec p vec q vec r])/(x^2) has least value =(3/2)^(2//3) c. [ vec p vec q vec r]>0 d. none of these

( vec a+ vec b)dot( vec b+ vec c)xx( vec a+ vec b+ vec c)= a. [ vec a\ vec b\ vec c] b. 0 c. 2[ vec a\ vec b\ vec c] d. -[ vec a\ vec b\ vec c]

If vec a , vec b , vec ca n d vec d are the position vectors of the vertices of a cyclic quadrilateral A B C D , prove that (| vec axx vec b+ vec bxx vec d+ vec d xx vec a|)/((vec b- vec a).(vec d- vec a)) + (| vec bxx vec c+ vec cxx vec d+ vec d xx vec b|)/((vec b- vec c).( vec d- vec c))=0dot

The length of the perpendicular form the origin to the plane passing through the point a and containing the line vec r= vec b+lambda vec c is a. ([ vec a vec b vec c])/(| vec axx vec b+ vec bxx vec c+ vec cxx vec a|) b. ([ vec a vec b vec c])/(| vec axx vec b+ vec bxx vec c|) c. ([ vec a vec b vec c])/(| vec bxx vec c+ vec cxx vec a|) d. ([ vec a vec b vec c])/(| vec cxx vec a+ vec axx vec b|)

Given vec a=x hat i+y hat j+2 hat k , vec b= hat i- hat j+ hat k , vec c= hat i+2 hat j ; vec a_|_ vec b , vec adot vec c=4. Then [ vec a vec b vec c]^2=| vec a| b. [ vec a vec b vec c]^=| vec a| c. [ vec a vec b vec c]^=0 d. [ vec a vec b vec c]^=| vec a|^2