Value of `[ vec axx vec b vec axx vec c vec d]` is always equal to `( vec a . vec d)[ vec a vec b vec c]` b. `( vec a . vec c)[ vec a vec b vec d]` c. `( vec a . vec b)[ vec a vec b vec d]` d. none of these

If [ vec a vec b vec c]=2, then find the value of [( vec a+2 vec b- vec c)( vec a- vec b)( vec a- vec b- vec c)]dot

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

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

If vec x+ vec cxx vec y= vec a and vec y+ vec cxx vec x= vec b ,where vec c is a nonzero vector, then which of the following is not correct? a. vec x=( vec bxx vec c+ vec a+( vec c . vec a) vec c)/(1+ vec c . vec c) b. vec x=( vec cxx vec b+ vec b+( vec c . vec a) vec c)/(1+ vec c . vec c) c. vec y=( vec axx vec c+ vec b+( vec c . vec b) vec c)/(1+ vec c . vec c) d. none of these

If vec a and vec b are orthogonal unit vectors, then for a vector vec r noncoplanar with vec a and vec b , vector rxxa is equal to a. [ vec r vec a vec b] vec b-( vec r. vec b)( vec bxx vec a) b. [ vec r vec a vec b]( vec a+ vec b) c. [ vec r vec a vec b] vec a-( vec r. vec a) vec axx vec b d. none of these

The condition for equations vec rxx vec a= vec ba n d vec rxx vec c= vec d to be consistent is a. vec b . vec c= vec a . vec d b. vec a . vec b= vec c .vec d c. vec b . vec c+ vec a . vec d=0 d. vec adot vec b+ vec c .vec d=0

The scalar vec Adot ( ( vec B+ vec C)xx( vec A+ vec B+ vec C)) equals a. 0 b. [ vec A vec B vec C]+[ vec B vec C vec A] c. [ vec A vec B vec C] d. none of these

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

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|)

If vec a is parallel to vec bxx vec c , then ( vec axx vec b)dot( vec axx vec c) is equal to a. | vec a|^2( vec b . vec c) b. | vec b|^2( vec a . vec c) c. | vec c|^2( vec a . vec b) d. none of these