If ` vec u , vec v and vec w` are three non-coplanar vectors, then prove that `( vec u+ vec v- vec w) . [ [( vec u- vec v)xx( vec v- vec w)]]= vec u . (vec v xx vec w)`

If vec a , vec b and vec c are three non-coplanar vectors, then ( vec a+ vec b+ vec c).[( vec a+ vec b)xx( vec a+ vec c)] equals a. 0 b. [ vec a vec b vec c] c. 2[ vec a vec b vec c] d. -[ vec a vec b vec c]

If vec a , vec b ,a n d vec c are three non-coplanar vectors, then find the value of ( vec a .( vec bxx vec c))/( vec b .( vec cxx vec a))+( vec b .( vec cxx vec a))/( vec c .( vec axx vec b))+( vec c . ( vec bxx vec a))/( vec a . ( vec bxx vec c))

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

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

If vec a , vec b and vec c are three non-zero, non coplanar vector vec b_1= vec b-( vec b . vec a)/(| vec a|^2) vec a , vec c_1= vec c-( vec c . vec a)/(| vec a|^2) vec a+( vec b . vec c)/(| vec c|^2) vec b_1 , , c_2= vec c-( vec c . vec a)/(| vec a|^2) vec a-( vec b . vec c)/(| vec b_1|^2) , b_1, vec c_3= vec c-( vec c . vec a)/(| vec c|^2) vec a+( vec b . vec c)/(| vec c|^2) vec b_1 , vec c_4= vec c-( vec c . vec a)/(| vec c|^2) vec a=( vec b . vec c)/(| vec b|^2) vec b_1 then the set of orthogonal vectors is a. ( vec a , vec b_1, vec c_3) b. ( vec a , vec b_1, vec c_2) c. ( vec a , vec b_1, vec c_1) d. ( vec a , vec b_2, vec c_2)

Let vec a , vec ba n d vec c be three non-coplanar vecrors and vec r be any arbitrary vector. Then ( vec axx vec b)xx( vec rxx vec c)+( vec bxx vec c)xx( vec rxx vec a)+( vec cxx vec a)xx( vec rxx vec b) is always equal to [ vec a vec b vec c] vec r b. 2[ vec a vec b vec c] vec r c. 3[ vec a vec b vec c] vec r d. none of these

vec u , vec v and vec w are three non-coplanar unit vecrtors and alpha,beta and gamma are the angles between vec u and vec v , vec v and vec w ,a n d vec w and vec u , respectively, and vec x , vec y and vec z are unit vectors along the bisectors of the angles alpha,betaa n dgamma , respectively. Prove that [ vecx xx vec y vec yxx vec z vec zxx vec x]=1/(16)[ vec u vec v vec w]^2sec^2(alpha/2)sec^2(beta/2)sec^2(gamma/2) .

If vec a and vec b are two vectors, then prove that ( vec axx vec b)^2=|[vec a.vec a,vec a.vec b],[vec b.vec a,vec b.vec b]| .

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

[( 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]