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)`

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) .

Show that the centroid of the triangle with vertices A(x_1,\ y_1,\ z_1) , B(x_2,\ y_2,\ z_2) and A(x_3,\ y_3,\ z_3) has the coordinates (("x"_1+"x"_2+"x"_2)/3,("y"_1+"y"_2+"y"_2,)/3("z"_1+"z"_2+"z"_2)/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)

A tetrahedron is a three dimensional figure bounded by four non coplanar triangular plane.So a tetrahedron has four no coplnar points as its vertices. Suppose a tetrahedron has points A,B,C,D as its vertices which have coordinates (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) respectively in a rectangular three dimensional space. Then the coordinates of its centroid are ((x_1+x_2+x_3+x_3+x_4)/4, (y_1+y_2+y_3+y_3+y_4)/4, (z_1+z_2+z_3+z_3+z_4)/4) . the circumcentre of the tetrahedron is the center of a sphere passing through its vertices. So, this is a point equidistant from each of the vertices of the tetrahedron. Let a tetrahedron have three of its vertices represented by the points (0,0,0) ,(6,-5,-1) and (-4,1,3) and its centroid lies at the point (1,2,5). The coordinate of the fourth vertex of the tetrahedron is

Show that the following systems of linear equations are inconsistent: 3x-y+2z=3,\ \ 2x+y+3z=5,\ \ x-2y-z=1

A plane meets the coordinate axes at P, Q and R such that the centroid of the triangle is (3,3,3). The equation of he plane is (A) x+y+z=9 (B) x+y+z=1 (C) x+y+z=3 (D) 3x+3y+3z=1

Let x_1y_1z_1,x_2y_2z_2 and x_3y_3z_3 be three 3-digit even numbers and Delta=|{:(x_1,y_1,z_1),(x_2,y_2,z_2),(x_3,y_3,z_3):}| . then Delta is

The length of projection of the segment join (x_1,y_1,z_1) and (x_2,y_2,z_20 on te line (x-alpha)/l=(y-beta)/m=(z-gamma)/n is (A) |l(x_2-x_1)+m(y_2-y_1)+n(z_2-z_1) (B) |alpha(x_2-x_1)+beta(y_2-y_1)+gamma(z_2-z_1)| (C) |(x_2-x_1)/l+(y_2-y_1)/m+(z_2-z_1)/n| (D) none of these

Show that the following system of equations is consistent. x-y+z=3,2x+y-z=2,-x-2y+2z=1

If the points A(3-x, 3, 3), B(3, 3-y, 3), C(3, 3, 3-z) and D(2, 2, 2) are coplanar, then (1)/(x)+(1)/(y)+(1)/(z) is equal to