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 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

If vec a , vec ba n d vec c are three non-coplanar vectors, then ( vec a+ vec b+ vec c)dot[( 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]

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 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

If vec r . vec a= vec r . vec b= vec rdot vec c=1/2 or some nonzero vector vec r , then the area of the triangle whose vertices are A( vec a),B( vec b)a n dC( vec c)i s( vec a , vec b , vec c are non-coplanar ) a. |[ vec a vec b vec c]| b. | vec r| c. |[ vec a vec b vec c] vec r| 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

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

vec a .(vec b + vec c) is equal to

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

The lines vec r= vec a+lambda( vec bxx vec c)a n d vec r= vec b+mu( vec cxx vec a) will intersect if a. vec axx vec c= vec bxx vec c b. vec adot vec c= vec bdot vec c c. bxx vec a= vec cxx vec a d. none of these