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]|`

The determinant |(b_1+c_1,c_1+a_1,a_1+b_1),(b_2+c_2,c_2+a_2,a_2+b_2),(b_3+c_3,c_3+a_3,a_3+b_3)|=

If A_1, B_1, C_1,... are respectively the co-factors of the elements a_1,b_1, c_1,... of the determinant Delta=|[a_1, b_1,c_1] , [a_2, b_2, c_2] , [a_3, b_3,c_3]| then |[B_2,C_2] , [B_3,C_3]|=

Evaluate: /_\ |[1+a_1, a_2, a_3],[a_1, 1+a_2, a_3],[a_1, a_2, 1+a_3]|

If Delta=|[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]| and A_1,B_1,C_1 denote the co-factors of a_1, b_1, c_1 respectively, then the value of the determinant |[A_1,B_1,C_1],[A_2,B_2,C_2],[A_3,B_3,C_3]| is-

Let veca = a_1hati + a_2hatj + a_3hatk, vecb = b_1hati + b_2hatj+ b_3hatk and vecc = c_1hati + c_2hatj + c_3hatk be three non zero vectors such that |vecc| =1 angle between veca and vecb is pi/4 and vecc is perpendicular to veca and vecb then |[a_1, b_1, c_1], [a_2, b_2, c_2], [a_3, b_3, c_3]|^2= lamda(a_1 ^2 +a_2 ^2 + a_3 ^2)(b_1 ^2 + b_2^2+b_3^2) where lamda is equal to (A) 1/2 (B) 1/4 (C) 1 (D) 2

Let vec a = a_1 hat i + a_2 hat j+ a_3 hat k;vec b = b_1 hat i+ b_2 hat j+ b_3 hat k ; vec c= c_1hat i + c_2 hat j+ c_3 hat k be three non-zero vectors such that vec c is a unit vector perpendicular to both vec a & vec b. If the angle between vec a and vec b is pi/6 , then |(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|^2=

If (b_2-b_1)(b_3-b-1)+(a_2-a_1)(a_3-a_1)=0 , then prove that the circumcenter of the triangle having vertices (a_1,b_1),(a_2,b_2) and (a_3,b_3) is ((a_2+a_3)/(2),(b_2+b_3)/(2)) .

If a_mhati+b_mhatj+c_mhatk m=1,2,3 are pairwise perpendicular unit vectors then |[a_1,b_1,c_1] , [a_2,b_2,c_2] , [a_3,b_3,c_3]| is equal to