Prove that `vec(a)[(vec(b) + vec(c)) xx (vec(a) + 3vec(b) + 4vec(c))] = [ vec(a) vec(b) vec(c)]`

Prove that vec(a). {(vec(b) + vec(c)) xx (vec(a) + 2vec(b) + 3vec(c))} = [vec(a) vec(b) vec(c)] .

If vec(a)vec(b)vec(c ) and vec(d) are vectors such that vec(a) xx vec(b) = vec(c ) xx vec(d) and vec(a) xx vec(c ) = vec(b)xxvec(d) . Then show that the vectors vec(a) -vec(d) and vec(b) -vec(c ) are parallel.

If vec(a), vec(b) and vec(c) are three vectors such that vec(a) times vec(b)=vec(c) and vec(b) times vec(c)=vec(a)," prove that "vec(a), vec(b), vec(c) are mutually perpendicular and abs(vec(b))=1 and abs(vec(c))=abs(vec(a)) .

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

Let vec(a), vec(b), vec(c) be vectors of length 3, 4, 5 respectively. Let vec(a) be perpendicular to vec(b)+vec(c), vec(b) to vec(c)+vec(a) and vec(c) to vec(a)+vec(b) . Then |vec(a)+vec(b)+vec(c)| is :

Prove that: ( vec a- vec b)*{( vec b- vec c)xx(vec c- vec a)}"=0

If the position vector of these points are vec(a) -2vec(b)+3 vec(c ), 2 vec(a)+3vec(b)-4 vec( c) ,-7 vec(b) + 10 vec(c ) , then the three points are

Prove that ( vec a- vec b)dot{( vec b- vec c)xx( vec c- vec a)}=0

If vec(a) , vec( b) and vec( c ) are unit vectors such that vec(a) + vec(b) + vec( c) = 0 , then the values of vec(a). vec(b)+ vec(b) . vec( c )+ vec( c) .vec(a) is

Show that vec(a), vec(b), vec(c ) are coplanar if and only if vec(a) + vec(b), vec(b) + vec(c ), vec( c) + vec(a) are coplanar