If ` vec u , vec va n d vec w` are three non-cop0lanar vectors, then prove that `( vec u+ vec v- vec w)dot( vec u- vec v)xx( vec v- vec w)= vec udot vec vdotxx vec w`

If vec a , vec ba n d vec c are three non coplanar vectors, then prove that vec d=( vec adot vec d)/([ vec a vec b vec c])( vec bxx vec c)+( vec bdot vec d)/([ vec a vec b vec c])( vec cxx vec a)+( vec cdot vec d)/([ vec a vec b vec c])( vec axx vec b)

Prove that if [ vec l vec m vec n] are three non-coplanar vectors, then [ vec l vec m vec n]( vec axx vec b)=| vec ldot vec a vec ldot vec b vec l vec mdot vec a vec mdot vec b vec m vec ndot vec a vec ndot vec b vec n| .

If a ,ba n dc are non-cop0lanar vector, then that prove |( vec adot vec d)( vec bxx vec c)+( vec bdot vec d)( vec cxx vec a)+( vecc dot vec d)( vec axx vec b)| is independent of d ,w h e r ee is a unit vector.

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

If vec a,vec b and vec c are three non-zero vectors,prove that [vec a+vec b,vec b+vec c,vec c+vec a]=2[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

If vec a , vec b , vec c are three non coplanar vectors such that vec ddot vec a= vec ddot vec b= vec ddot vec c=0, then show that d is the null vector.

If vec a, vec b , vec c are three non- coplanar vectors such that vec a + vec b + vec c = alpha vec d and vec b +vec c + vec d = beta vec a, " then " vec a + vec b + vec c + vec d to equal to

Let vec u , vec va n d vec w be vectors such that vec u+ vec v+ vec w=0. If | vec u|=3,| vec v|=4a n d| vec w|=5, then vec udot vec v+ vec vdot vec w+ vec wdot vec u is 47 b. -25 c. 0 d. 25