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)`.

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 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)

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)

The value of the determinant |{:(1,x,x^2),(1,y,y^2),(1,z,z^2):}| is equal to

The value of the determinant |{:(1,x,x^2),(1,y,y^2),(1,z,z^2):}| is equal to

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

Prove that : =|{:(1,1,1),(x,y,z),(x^(2),y^(2),z^(2)):}|=(x-y)(y-z)(z-x)

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

If z_1=x_1+iy_1,z_2=x_2+iy_2 and z_1 = (i(z_2+1))/(z_2-1) , prove that x_1^2+y_1^2-x_1= (x_2^2+y_2^2+2x_2-2y_2+1)/((x_2-1)^2+y_2^2)