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 zero: (a)|{:(a_(1),,la_(1)+mb_(1),,b_(1)),(a_(2),,la_(2)+mb_(2),,b_(2)),( a_(3),,la_(3)+mb_(3),,b_(3)):}|

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)

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

If w is a complex cube root of unity, then value of =|a_1+b_1w a_1w^2+b_1c_1+b_1 w a_2+b_2w a_2w^2+b_2c_2+b_2 w a_3+b_3w a_3w^2+b_3c_3+b_3 w | is a. 0 b. -1 c. 2 d. none of these

Find the first five terms of the following sequence . a_1 = 1 , a_2 = 1 , a_n = (a_(n-1))/(a_(n-2) + 3) , n ge 3 , n in N

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| is a. 0 b. 1 c. 1/4(a_1 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)

If A_1B_1C_1 , A_2B_2C_2 and A_3B_3C_3 are three digit numbers, each of which is divisible by k, then Delta = |(A_1,B_1,C_1),(A_2,B_2,C_2),(A_3,B_3,C_3)| is

Suppose four distinct positive numbers a_1, a_2, a_3, a_4, are in G.P. Let b_1=a_1,b_2=b_1+a_2.b_3=b_2+a_3 and b_4=b_3+a_1.