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]|`

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

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

Consider the determinant Delta = |[a_1+b_1x^2,a_1x^2+b_1,c_1],[a_2+b_2x^2,a_2x^2+b_2,c_2],[a_3+b_3x^2,a_3x^2+b_3,c_3]| = 0 , \ w h e r e \ a_i ,b_i , c_i in R \ (i = 1,2,3) \ a n d \ x in R . Statement 1: The value of x satisfying Delta=0 are x=1,-1. Statement 2: If |[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]|=0,t h e n \ Delta=0.

If w is a complex cube root of unity, then value of =|[a_1+b_1w, a_1w^2+b_1,c_1+b_1 w],[ a_2+b_2w, a_2w^2+b_2,c_2+b_2 w],[ a_3+b_3w ,a_3w^2+b_3,c_3+b_3 w] | is a. 0 b. -1 c. 2 d. none of these

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=

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

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 (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)

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 and vec c=c_1 hat 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 and vec b . If the angle between a and b is pi/6, then prove that |[a_1,a_2,a_3],[b_1,b_2,b_3],[c_1,c_2,c_3]|^2=1/4(a_1 ^2+a_2 ^2+a_3 ^2)(b_1 ^2+b_2 ^2+b_3 ^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 ka n d vec c=c_1 hat i+c_2 hat j+c_3 hat k be three nonzero vectors such that vec c is a unit vector perpendicular to both vec aa n d vec bdot If the angle between vec aa n d vec b is pi//6 , then the value of |a_1b_1c_1a_2b_2c_2a_3b_3c_3|^2 is a. 0 b. 1 c. 1/4(a1^2+a2^2+a3^2)(b1^2+b2^2+b3^2) d. 3/4(a1^2+a2^2+a3^2)(b1^2+b2^2+b3^2)