Let the pairs `a , ba n dc ,d` each determine a plane. Then the planes are parallel if `( vec axx vec c)xx( vec bxx vec d)= vec0` b. `( vec axx vec c)dot( vec bxx vec d)= vec0` c. `( vec axx vec b)xx( vec cxx vec d)= vec0` d. `( vec axx vec b)dot( vec cxx vec d)= vec0`

The pairs vec a,vec b and vec c,vec d each determines plane.Then the planes are parallel if

[vec a, vec b + vec c, vec d] = [vec a, vec b, vec d] + [vec a, vec c, vec d]

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)

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)

If [vec b vec c vec d] = 24 and (vec a xx vec b) xx (vec c xx vec d) + (vec a xx vec c) xx (vec d xx vec b) + (vec a xx vec d ) xx (vec b xx vec c) + k vec a = 0 then k is equal to

vec a, vec b, vec c, dare any four vectors then (vec a xxvec b) xx (vec c xxvec d) is a vector Perpendicular to vec a, vec b, vec c, vec d

For any four vectors,vec a,vec b,vec c and vec d prove that vec d*(vec a xx(vec b xx(vec c xxvec d)))=(vec b*vec d)[vec avec cvec d]

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|)

The vector component of vec b perpendicular to vec a is ( vec bdot vec c) vec a b. ( vec axx( vec bxx vec a))/(| vec a|^2) c. vec axx( vec bxx vec a) d. none of these

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 xuuR^+, then x[ vec a vec b vec c]+([ vec p vec q vec r])/x 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