Prove that the normal to the plane containing three points whose position vectors are ` vec a , vec b , vec c` lies in the direction ` vec bxx vec c+vec cxx vec a+ vec axx vec bdot`

(a) Prove that the normal to the plane containing three points whose position vectors are vec(a), vec(b), vec(c ) lie in the direction of vec(b)xx vec(c )+vec(c )xx vec(a)+vec(a)xx vec(b) . (b) Find the unit vector perpendicular to the plane ABC, where the position vectors of A, B and C are : 2hat(i)-hat(j)+hat(k), hat(i)+hat(j)+2hat(k) and 2hat(i)+hat(k) respectively.

Show that the points whose position vectors are vec(a) + 2vec(b) + 5vec(c), 3vec(a) + 2vec(b) + vec(c), 2vec(a) + 2vec(b) + 3vec(c) are colliner.

Prove that the points A,B and C with position vectors vec a,vec b and vec c respectively are collinear if and only if vec a xxvec b+vec b xxvec c+vec c xxvec a=vec 0

For any three vectors vec a, vec b, vec c, (vec a-vec b) * (vec b-vec c) xx (vec c-vec a) is equal to

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,vec b and vec c be any three vectors then show that vec a+(vec b+vec c)=(vec a+vec b)+vec c

For any three vectors vec a;vec b;vec c find [vec a+vec b;vec b+vec c;vec c+vec a]

Show that the points with position vectors vec a-2vec b+3vec c,-2vec a+3vec b-vec c and 4vec a-7vec b+7vec c are collinear.

If O A B C is a tetrahedron where O is the orogin anf A ,B ,a n dC are the other three vertices with position vectors, vec a , vec b ,a n d vec c respectively, then prove that the centre of the sphere circumscribing the tetrahedron is given by position vector (a^2( vec bxx vec c)+b^2( vec cxx vec a)+c^2( vec axx vec b))/(2[ vec a vec b vec c]) .

Prove that,for any three vectors vec a,vec b,vec c[vec a+vec b,vec b+vec c,vec c+vec a]=2[vec a,vec b,vec c]