Let `a=xhati+12hatj-hatk,b=2hati+2xhatjj+hatk and c=hati+hatk`. If b,c,a in that order form a left handed system, then find the value of x.
`[x_(1)a+y_(1)b+z_(1)c,x_(2)a+y_(2)b+z_(2)c,x_(3)a+y_(3)b+z_(3)c]`
`=|(x_(1),y_(1),z_(1)),(x_(2),y_(2),z_(2)),(x_(3),y_(3),z_(3))|[abc]`.

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)]]

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))

Consider the system of equations a_(1) x + b_(1) y + c_(1) z = 0 a_(2) x + b_(2) y + c_(2) z = 0 a_(3) x + b_(3) y + c_(3) z = 0 If |(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| =0 , then the system has

Cosnsider the system of equation a_(1)x+b_(1)y+c_(1)z=0, a_(2)x+b_(2)y+c_(2)z=0, a_(3)x+b_(3)y+c_(3)z=0 if |{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=0 , then the system has

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))

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))]]

If veca=x hati+y hatj+z hatk, vecb=y hati+z hatj+x hatk, and vec c=z hati+x hatj+y hatk , then veca xx (vecb xx vec c) is :

If x y z=0, then find the value of (a^x)^(y z)+(a^y)^(z x)+(a^z)^(x y)= (a)3 (b) 2 (c)1 (d) 0

Sow that x_(1)hat i+y_(1)hat j+z_(1)hat k,x_(2)hat i+y_(2)hat j+z_(2)hat k, and x_(3)hat i+y_(3)hat j+z_(3)hat k are non-coplanar if |x_(1)|>|y_(1)|+|z_(1)|,|y_(2)|>|x_(2)|+|z_(2)| and |z_(3)|>|x_(3)|+|y_(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)