Let `a=xhati+12hatj-hatk,b=2hati+2xhatj+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]`.

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

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

proof |[x,y,z],[x^(2),y^(2),z^(2)],[yz,zx,xy]| = |[1,1,1],[x^(2),y^(2),z^(2)],[x^(3),y^(3),z^(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)

Evaluate |{:(.^(x)C_(1),,.^(x)C_(2),,.^(x)C_(3)),(.^(y)C_(1),,.^(y)C_(2),,.^(y)C_(3)),(.^(x)C_(1),,.^(z)C_(2),,.^(z)C_(3)):}|

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

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)a n d\ (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)

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 :

If vec r = 3 hati + 2 hatj - 5 hatk , vec a= 2 hati - hatj + hatk, vec b = hati + 3 hatj - 2hatk and vec c=-2 hati + hatj - 3 hatk " such that " hat r = x vec a +y vec b + z vec c then

if x,y and z are not all zero and connected by the equations a_(1)x+b_(1)y+c_(1)z=0,a_(z)x+b_(2)y+c_(2)z=0 and (p_(1)+lambdaq_(1))x+(p_(2)+lambdaq_(2))y+(p_(3)+lambdaq_(3))z=0 show that lambda =-|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(p_(1) ,, p_(2),,p_(3)):}|-:|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(q_(1) ,, q_(2),,q_(3)):}|