Show that `vec(a)xx vec(b)=vec(a)xx vec(c )` does not imply `vec(b)=vec(c )`. Illustrate geometrically.

If vec(a), vec(b), vec(c ) are three vectors such that vec(a)xx vec(b)=vec(c ), vec(b)xx vec(c )=vec(a) , prove that vec(a), vec(b), vec(c ) are mutually at right angles and |vec(b)|=1, |vec(c )|=|vec(a)| .

If vec(a)xx vec(b)=vec(b)xx vec(c )ne vec(0) , show that vec(a)+vec(c )=vec(mb) , m being a scalar.

Show that (vec(A) - vec(B)) xx (vec(A) +vec(B)) = 2(vec(A) xx vec(B))

The vectors vec(a), vec(b), vec(c ) and vec(d) are such that vec(a) xx vec(b) = vec(c ) xx vec(d) and vec(a) xx vec(c )= vec(b) xx vec(d) . Which of the following is/are correct? 1. (vec(a)- vec(d)) xx (vec(b) - vec(c ))= vec(0) 2. (vec(a) xx vec(b))xx (vec(c ) xx vec(d))= vec(0) Select the correct answer using the codes given below

If vec(a), vec(b), vec(c ) are three vectors such that vec(a)+vec(b)+vec(c )=0 , then prove that : vec(a)xx vec(b)=vec(b)xx vec(c )=vec(c )xx vec(a) and hence, show that [vec(a)vec(b)vec(c )]=0 .

The vectors vec(a), vec(b), vec(c) and vec(d) are such that vec(a)xx vec(b) = vec(c) xx d and vec(a) xx vec(c)= vec(b) xx vec(d) . Which of the following is/are correct? 1. (vec(a)-vec(d))xx (vec(b)-vec(c))=vec(0) 2. (vec(a)xx vec(b))xx (vec(c)xx vec(d))=vec(0) Select the correct answer using the code given below :

if vec(a) + 2vec(b) + 3vec(c ) = 0 and vec(a) xx vec(b) + vec(b) xx vec(c ) + vec(c ) xx vec(a) = lamda (vec(b) xx vec(c )) then what is the value of lamda ?

If vec(a), vec(b), vec(c ) are position vectors of non - collinear points A, B and C respectively, show that : vec(a)xx vec(b)+vec(b)xx vec(c )+vec(c )xx vec(a) is perpendicular to the plane ABC.

Show that if vec(a)+vec(b), vec(b)+vec(c ), vec(c )+vec(a) are coplanar, then vec(a), vec(b), vec(c ) are also coplanar.

If vec(a),vec(b),vec(c) are three non-coplanar unit vector such that [vec(a),vec(b),vec(c)]=3 then [ vec(a) times vec(b), vec(b) times vec (c),vec(c) times vec(a) ] equals