Find the values of x,y for following question `(a_3-a_2)x+(b_3-b_2)y`=`((a_2^2+b_2^2)-(a_3^2+b_3^2))/2` and `(a_3-a_1)x+(b_3-b_1)y=((a_1^2+b_1^2)-(a_3^2+b_3^2))/2`

The value of the determinant (a_1-b_1)^2(a_1- b_2)^2(a_1-b_3)^2(a_1-b_4)^2(a_2-b_1)^2(a_2-b_2)^2(a_2-b_3)^2(a_2- b_4)^2(a_3-b_1)^2(a_3-b_2)^2(a_3-b_3)^2(a_3-b_4)^2(a_4-b_1)^2(a_4-b_2)^2(a_4-b_3)^2(a_4-b_4)^2 is dependant on a_i , i=1,2,3,4 dependant on b_i , i=1,2,3,4 dependant on a_(i j), b_i i=1,2,3,4

Find the coefficient of x in the determinant |((1+x)^(a_1 b_1),(1+x)^(a_1 b_2),(1+x)^(a_1 b_3)),((1+x)^(a_2 b_2),(1+x)^(a_2 b_2),(1+x)^(a_2 b_3)),((1+x)^(a_3 b_3),(1+x)^(a_3 b_2),(1+x)^(a_3 b_3))|=0.

Prove that the value of each the following determinants is zero: |[a_1,l a_1+mb_1,b_1],[a_2,l a_2+mb_2,b_2],[a_3,l a_3+m b_3,b_3]|

Prove that the value of each the following determinants is zero: |[a_1,l a_1+mb_1,b_1],[a_2,l a_2+mb_2,b_2],[a_3,l a_3+m b_3,b_3]|

Prove that the value of each the following determinants is zero: |[a_1,l a_1+mb_1,b_1],[a_2,l a_2+mb_2,b_2],[a_3,l a_3+m b_3,b_3]|

Prove that the value of each the following determinants is zero: |[a_1,l a_1+mb_1,b_1],[a_2,l a_2+mb_2,b_2],[a_3,l a_3+m b_3,b_3]|

Prove by vector method that (a_1b_1+a_2b_2+a_3b_3)^2lt+(a_1^2+a_2^2+a_3^2)(b_1^2+b_2^2+b_3^2)

Find the coordinates of the centroid of the triangle with its vertices at (a_1,b_1,c_1),(a_2,b_2,c_2), "and"(a_3, b_3,c_3).