R D Sharma Solutions for Chapter: Scalar or Dot Product, Exercise 2: EXERCISE

Prove that: If the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

Prove by vector method that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.

Prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.

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

Prove that the diagonals of a rectangle are perpendicular if and only if the rectangle is a square.

If $AD$ is the median of $∆ABC$, using vectors, prove that $A{B}^2+A{C}^2=2\left(A{D}^2+C{D}^2\right)$.

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

In a quadrilateral $ABCD$, prove that $A{B}^2+B{C}^2+C{D}^2+D{A}^2=A{C}^2+B{D}^2+4P{Q}^2$where $P$ and $Q$ are middle points of diagonals $AC$ and $BD$.