Three Points With Position Vectors  will Be Collinear, If There Exist Scalars x, y, z such That   

If O and O’ denote respectively the circum-centre and orthocentre of∆ABC, then 

If O and O’ denote respectively the circum-centre and orthocenter of âˆ†ABC, then  

Consider âˆ†ABC and ∆A₁B₁C₁ in such a way that  and M, N, M₁, N₁ be the mid-point of AB, BC, A₁B₁ and B₁C₁ respectively. Then,  

Let ABCD be a parallelogram whose diagonals intersect at P and let O be the origin, then  equals

If A, B, C, D be any four points and E and F be the middle points of AC and BD respectively, then  is equal to

Given that the vectors  are non-collinear, the values of x and yfor which the vector equality  holds true if  are  

Let  be three non-zero vectors, no two of which are collinear. If the vector  is collinear with  is collinear with  is equal to 

If the position vector of the three points are ,

 then the three points are 

The position vectors of the vertices A, B, C of a âˆ†ABC are  respectively. The length of the bisector AD of the angle ∠BAC where D is on the line segment BC, is