Section Formula in Coordinate Geometry

If `bara, barb` and `barr` are position vectors of the points A, B and R respectively and R divides the line segment AB externally in the ratio m : n, then prove that `barr = (mbarb - nbara)/(m - n)`.

The co-ordinates of the points which divides line segment joining the point A(2, –6, 8) and B(–1, 3,–4) internally in the ratio 1: 3' are ______.

Let `A(bara)` and `B(barb)` are any two points in the space and `R(barr)` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `barr = (mbarb + nbara)/(m + n)`.

If the median to the base of a triangle is perpendicular to the base, then triangle is isosceles. 

Find the position vector of point R which divides the line joining the points P and Q whose position vectors are `2hat"i" - hat"j" + 3hat"k"` and  `- 5hat"i" + 2hat"j" - 5hat"k"` in the ratio 3 : 2 is externally.

Prove that the diagonals of a rhombus are perpendicular bisectors of each other. 

Find the coordinate of the point P where the line through A(3, –4, –5) and B(2, –3, 1) crosses the plane passing through three points L(2, 2, 1), M(3, 0, 1) and N(4, –1, 0).
Also, find the ratio in which P divides the line segment AB.

The position vector of points A and B are `6 bara + 2 barb and bara - 3 barb`. If the point C divides AB in the ratio 3 : 2 then show that the position vector of C is `3 bara - barb`.