If `O` be the origin, and if the coordinates of any two points `P_1`, and `P_2`, be respectively `(x_1, y_1)` and `(x_2,y_2),` prove that: `OP_1. OP_2 .cos P_1OP_2 = x_1.x_2 + y_1.y_2.`

If O be the origin,and if the coordinates of any two points P_(1), and P_(2), be respectively (x_(1),y_(1)) and (x_(2),y_(2)), prove that: OP_(1).OP_(2).cos P_(1)OP_(2)=x_(1)*x_(2)+y_(1)*y_(2)

If O is the origin and if the coordinates of any two points Q_(1) and Q_(2) are (x_(1),y_(1)) and (x_(2),y_(2)), respectively,prove that OQ_(1).OQ_(2)cos/_Q_(1)OQ_(2)=x_(1)x_(2)+y_(1)y_(2)

Find the coordinates of the centroid of a triangle having vertices P(x_1,y_1,z_1),Q(x_2,y_2,z_2) and R(x_3,y_3,z_3)

Three points A(x_1 , y_1), B (x_2, y_2) and C(x, y) are collinear. Prove that: (x-x_1) (y_2 - y_1) = (x_2 - x_1) (y-y_1) .

If the line segment joining the points P(x_1, y_1)a n d Q(x_2, y_2) subtends an angle alpha at the origin O, prove that : O PdotO Q cosalpha=x_1x_2+y_1y_2dot

If O is the origin,OP=3 with direction ratios -1,2, and -2, then find the coordinates of P.

The ratio in which the line segment joining P(x_1, y_1) and Q(x_2, y_2) is divided by x-axis is y_1: y_2 (b) y_1: y_2 (c) x_1: x_2 (d) x_1: x_2

If O be the origin and the coordinates of P be(1," "2," "" "-3) , then find the equation of the plane passing through P and perpendicular to OP.

If O be the origin and the coordinates of P be (1,2,-3) then find the equation of of the plane passing through P and perpendicular to OP.