Prove that `|[a, a+b, a+b+c],[ 2 a, 3 a+2 b, 4 a+3 b+2 c],[ 3 a, 6 a+3 b, 10 a+6 b+3 c]|=a^3`

Consider abs[[a,a+b,a+b+c],[2a,3a+2b,4a+3b+2c],[3a,6a+3b,10a+6b+3c]]

Prove that |[1 , a , a^3],[ 1, b, b^3],[ 1, c, c^3]|=(a-b)(b-c)(c-a)(a+b+c)

Ifa,b,c are vectors such that a + b + c = 0 and | a | = 7, | b | = 5,| c | = 3, then the angle between c and b is a) pi//3 b) pi//6 c) pi//4 d) pi

If the system of equations x - 2y +z = a , 2x + y - 2z = b, x +3y - 3z = c, has at least one solution, then a) a + b + c = 0 b) a - b + c = 0 c) -a + b + c = 0 d) a + b - c = 0

If |(1,1,1),(a,b,c),(a^(3),b^(3),c^(3))| = (a - b) (b - c) (c - a) (a + b + c) , where a,b,c are all different, then the determinant |(1,1,1),((x-a)^(2),(x-b)^(2),(x-c)^(2)),((x-b)(x-c),(x-c)(x-a),(x-a)(x-b))| vanishes when a)a + b + c = 0 b) x = (1)/(3) (a + b + c) c) x = (1)/(2) (a + b + c) d) x = a + b + c

If A =[[1, 2, -3],[ 5, 0, 2],[ 1, -1, 1]], B=[[3, -1, 2],[ 4, 2, 5],[ 2, 0, 3]] and C=[[4 ,1 , 2],[ 0, 3, 2],[ 1, -2, 3]] then compute (A+B) and (B-C) . Also verify that A+(B-C)=(A+B)-C