If a, b, c and a', b', c' are recoprocal system of vectors, then prove that `a'timesb'+b'timesc'+c'timesa'=(a+b+c)/([a b c])`.

If a, b and c are three non-coplanar vectors, then find the value of (a*(btimesc))/(ctimes(a*b))+(b*(ctimesa))/(c*(atimesb)) .

If a, b, c and d are four coplanr points, then prove that [a b c]=[b c d]+[a b d]+[c a d] .

If a,b and c are any three vectors in space, then show that (c+b)xx(c+a)*(c+b+a)= [a b c]

If a+b+c!=0 and |a b c b c a c a b|=0 , then prove that a=b=cdot

If a ,b ,c are positive, then prove that a//(b+c)+b//(c+a)+c//(a+b)geq3//2.

If a,b and c are non-coplanar vectors, prove that 3a-7b-4c, 3a-2b+c and a+b+2c are coplanar.

If A=2B , then prove that either c=b or a^2 = b(c+b)

vec a , vec b and vec c are the position vectors of points A ,B and C respectively, prove that : vec a× vec b+ vec b× vec c+ vec c× vec a is vector perpendicular to the plane of triangle A B Cdot

If a,b,c,l and m are vectors, prove that [a b c] (lxxm)=|(a,b,c),(a*l,b*l,c*l),(a*m,b*m,c*m)|

For any three vectors a,b and c prove that ["a b c"]^(2)=|(a*a,a*b,a*c),(b*a,b*b,b*c),(c*a,c*b,c*c)|