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))`.

Find the scalar components and magnitude of the vector joining the points P(x_(1),y_(1),z_(1))andQ(x_(2),y_(2),z_(2)) .

An equilateral triangle has each side equal to a. If the coordinates of its vertices are (x_(1), y_(1)), (x_(2), y_(2)) and (x_(3), y_(3)) then the square of the determinant |(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1)| equals

Find the circumcentre of the triangle whose vertices are given by the complex numbers z_(1),z_(2) and z_(3) .

The centroid of the triangle with verticies A(1, 1, 1), B(2, 1, 2) and C(x, y, z) is O(0, 0, 0) then (x, y, z) = ______ .

Show that the triangle whose vertices are z_(1)z_(2)z_(3)andz_(1)'z_(2)'z_(3)' are directly similar , if |{:(z_(1),z'_(1),1),(z_(2),z'_(2),1),(z_(3),z'_(3),1):}|=0

Complex numbers z_(1),z_(2),z_(3) are the vertices of A,B,C respectively of an equilteral triangle. Show that z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1).

If Y=sX and Z =tX all the varibles beings functions of x then prove that |{:(X,Y,Z),(X_(1),Y_(1),Z_(1)),(X_(2),Y_(2),Z_(2)):}| =X^(3) |{:(s_(1),t_(1)),(s_(2),t_(2)):}| where suffixes denote the order of differention with respect to x.

Find the equation of the image of the plane x-2y+2z-3=0 in plane x+y+z-1=0.

Find the image of the point (3, 5, 3) with respect to the line x/1=(y-1)/(2)=(z-2)/(3)