If `G` is the centroid and `O` is the circumcentre of the triangle with vertices `(1,2,0),(0,0,2)a n d(2,1,1)` , then equations of line `O G` are (a) `x=y=z` (b) `y=1,z=1` (c) `(x-1)/1=y/1=z/1` (d) `(x-1)/1=(y-1)/1=(z-1)/1`

Find the coordinates of the centroid of the triangle whose vertices are (x_1,y_1,z_1) , (x_2,y_2,z_2) and (x_3,y_3,z_3) .

If the volume of tetrahedron A B C D is 1 cubic units, where A(0,1,2),B(-1,2,1)a n dC(1,2,1), then the locus of point D is a. x+y-z=3 b. y+z=6 c. y+z=0 d. y+z=-3

The centroid of the triangle formed by (0, 0, 0) and the point of intersection of (x-1)/x=(y-1)/2=(z-1)/1 with x=0 and y=0 is

The equation of the line through (1,1,1) and perpendicular to the plane 2x+3y-z=5 is (A) (x-1)/2=(y-1)/3=z-1 (B) (x-1)/2=(y-1)/3=(z-1)/(-1) (C) (x-1)/2=(y-1)/3=(z-1)/(5) (D) (x-1)/2=(y-1)/(-3)=z-1

If the volume of tetrahedron A B C D is 1 cubic units, where A(0,1,2),B(-1,2,1) and C(1,2,1), then the locus of point D is a. x+y-z=3 b. y+z=6 c. y+z=0 d. y+z=-3

The lines (x+1)/(1)=(y-1)/(2)=(z-2)/(1),(x-1)/(2)=(y)/(1)=(z+1)/(4) are

Shortest distance between the lines (x-1)/1=(y-1)/1=(z-1)/1a n d(x-2)/1=(y-3)/1=(z-4)/1 is equal to

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)

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 area of the triangle having vertices (0,0,0),(x_1,y_1,z_1),(x_2,y_2,z_2) is 1/2 sqrt((y_1z_2-y_2z_1)^2+(z_2x_1-x_2z_1)^2+(x_1y_2-x_2y_1)^2) .