The value of `|(2 y_(1)z_(1),y_(1) z_(2) + y_(2) z_(1),y_(1) z_(3) + y_(3) z_(1)),(y_(1) z_(2) y_(2) z_(1),2y_(2) z_(2),y_(2) z_(3) + y_(3) z_(2)),(y_(1) z_(3) + y_(3) z_(1),y_(2) z_(3) + y_(3) z_(2),2y_(3) z_(3))|`, is

For x_(1), x_(2), y_(1), y_(2) in R if 0 lt x_(1)lt x_(2)lt y_(1) = y_(2) and z_(1) = x_(1) + i y_(1), z_(2) = x_(2)+ iy_(2) and z_(3) = (z_(1) + z_(2))//2, then z_(1) , z_(2) , z_(3) satisfy :

Evaluate det[[x_(C_(1)),x_(C_(2)),x_(C_(3))y_(C_(1)),y_(C_(2)),yC_(3)z_(C_(1)),z_(C_(2)),z_(C_(3))]]

A=[[2,0,00,2,00,0,2]] and B=[[x_(1),y_(1),z_(1)x_(2),y_(2),z_(2)x_(3),y_(3),z_(3)]]

ax_(1)^(2)+by_(1)^(2)+cz_(1)^(2)=ax_(2)^(2)+by_(2)^(2)+cz_(2)^(2)=ax_(3)^(2)+by_(3)^(2)+cz_(3)^(2)=d,ax_(2)^(3)+by_(2)y_(3)+cz_(2)z_(3)=ax_(3)x_(1)+by then prove that det[[x_(1),y_(1),z_(1)x_(2),y_(2),z_(2)x_(3),y_(3),z_(3)]]=(d-f){((d+2f))/(abc)}^(1/2)

The shortest distance between the lines (x-x_(1))/(a_(1)) =(y-y_(1))/(b_(1))=(z-z_(1))/(c_(1)) and (x-x_(2))/(a_(2))=(y-y_(2))/(b_(2))=(z-z_(2))/(c_(2)) is |{:(,x_(1)-x_(1),y_(2)-y_(1),z_(2)-z_(1)),(,a_(1),b_(1),c_(1)),(,a_(2),b_(2),c_(2)):}| /A Here, A refers to

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)

Show that the coordinates off the centroid of the triangle with vertices A(x_(1),y_(1),z_(1)),B(x_(2),y_(2),z_(2)) and (x_(3),y_(3),z_(3)) are ((x_(1)+x_(2)+x_(3))/(3),(y_(1)+y_(2)+y_(3))/(3),(z_(1)+z_(2)+z_(3))/(3))

Let x_(1) and y_(1) be real number. If z_(1) and z_(2) are complex numbers such that |z_(1)|=|z_(2)|=4 , then |x_(1)z_(1)-y_(1)z_(2)|^(2)+|y_(1)z_(1)+x_(1)z_(2)|^(2)=

Find the co oridinate of the centroid of the tetrahedron whose vertices are (x_(1),y_(1),z_(1)),(x_(2),y_(2),z_(2)),(x_(3),y_(3),z_(3)) and (x_(4),y_(4),z_(4))