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)|`

For any three unequal numbers a,b and c (a-b)/(b-c) equals to

If a ,b ,c ,d are in G.P. prove that: (a b-c d)/(b^2-c^2)=(a+c)/b

For any three vectors a ,\ b ,\ c prove that [ vec a+ vec b\ vec b+ vec c\ vec c+ vec a]=2\ [ vec a\ vec b\ vec c]dot

If a, b, c are three given non-coplanar vectors and any arbitratry vector r is in space, where Delta_1=|[r*a, b*a, c*a], [r*b, b*b, c*b], [r*c, b*c, c*c]| : Delta_2=|[a*a, r*a, c*a], [a*b, r*b, c*b], [a*c, r*c, c*c]| Delta_3=|[a*a, b*a, r*a], [a*b, b*b, r*b], [a*c, b*c, r*c]|, Delta_4=|[a*a, b*a, c*a], [a*b, b*b, c*b], [a*c, c*c, c*c]| Q. The vector r is expressible in the form

If a, b and c are three coplanar vectors. If a is not parallel to b, show that c=(|[c*a, a*b], [c*b, b*b]|a+|[a*a, c*a], [a*b, c*b]|b)/(|[a*a, a*b], [a*b, b*b]|) .

Prove that: |(-2a , a+b, a+c),( b+a,-2b, b+c),( c+a , c+b,-2c)|=4(a+b)(b+c)(c+a)

Prove that : |{:(a,b,c),(b,c,a),(c,a,b):}|=3 a b c-a^(3)-b^(3)-c^(3)

Prove that: |[-2a, a+b,a+c],[ b+a,-2b,b+c],[c+a, c+b,-2c]|=4(a+b)(b+c)(c+a)

Prove that: |[-2a, a+b,a+c],[ b+a,-2b,b+c],[c+a, c+b,-2c]|=4(a+b)(b+c)(c+a)

If a ,b ,c are in G.P. then prove that (a^2+a b+b^2)/(b c+c a+a b)=(b+a)/(c+b)