If `A=[(2,0,0),(0,2,0),(0,0,2)] and B=[(x_1,y_1,z_1),(x_2,y_2,z_2),(x_3,y_3,z_3)].` Prove that `AB=2B`

suppose that the mid point of the sides BC,CA and AB of a triangle Delta ABC are (5,7,11), (0,8,5) and (2,3,-1) If the vertices of the triangle are A(x_1,y_1,z_1), B(x_2,y_2,z_2) and C(x_3,y_3,z_3) derive nine equations in x_1,x_2,x_3,y_1,y_2,y_3,z_1,z_2, and z_3

If [(1,0,0),(0, 0, 1),(0,1,0)][(x),(y),(z)]=[(2),(-1),(3)] , find x ,\ y ,\ z .

If a x_1^2+b y_1^2+c z_1^2=a x_2 ^2+b y_2 ^2+c z_2 ^2=a x_3 ^2+b y_3 ^2+c z_3 ^2=d ,a x_2 x_3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1=a x_1x_2+b y_1y_2+c z_1z_2=f, then prove that |(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3,y_3,z_3)|=(d-f){((d+2f))/(a b c)}^(1//2)

If a x_1^2+b y_1^2+c z_1^2=a x_2 ^2+b y_2 ^2+c z_2 ^2=a x_3 ^2+b y_3 ^2+c z_3 ^2=d ,a x_2 x_3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1=a x_1x_2+b y_1y_2+c z_1z_2=f, then prove that |(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3,y_3,z_3)|=(d-f){((d+2f))/(a b c)}^(1//2)

STATEMENT-1 : The centroid of a tetrahedron with vertices (0, 0,0), (4, 0, 0), (0, -8, 0), (0, 0, 12)is (1, -2, 3). and STATEMENT-2 : The centroid of a triangle with vertices (x_(1), y_(1), z_(1)), (x_(2), y_(2), z_(2)) and (x_(3), y_(3), z_(3)) is ((x_(1)+x_(2)+x_(3))/3, (y_(1)+y_(2)+y_(3))/3, (z_(1)+z_(2)+z_(3))/3)

STATEMENT-1 : The centroid of a tetrahedron with vertices (0, 0,0), (4, 0, 0), (0, -8, 0), (0, 0, 12)is (1, -2, 3). and STATEMENT-2 : The centroid of a triangle with vertices (x_(1), y_(1), z_(1)), (x_(2), y_(2), z_(2)) and (x_(3), y_(3), z_(3)) is ((x_(1)+x_(2)+x_(3))/3, (y_(1)+y_(2)+y_(3))/3, (z_(1)+z_(2)+z_(3))/3)

Given X = {:[(2,0,-2),(1,0,-2)] and Y = [(3,-1,0),(-2,0,-1)] , find Z such that X+Y+Z = 0

If O(0, 0, 0), A(x, 1, -1), B(0, y, 2) and C(2, 3, z) are coplanar, then

If 2^x=3^y=6^(-z) prove that 1/x+1/y+1/z=0