Odisha Board Solutions for Chapter: Vectors, Exercise 2: EXERCISE-12(b)

If $\overrightarrow{a}·\overrightarrow{b}=\overrightarrow{c}·\overrightarrow{a}$ for all vector $\overrightarrow{a}$, then 

Prove by vector method, median to the base of an isosceles triangle is perpendicular to the base.

Prove by vector method that the parallelogram whose diagonals are equal is a rectangle.

Prove by vector method that the diagonals are at right angles.

Prove by vector method that an angle inscribed in a semicircle is a right angle.

Prove by vector method that in any triangle $ABC$, $a=b\cos C+c\cos B$.

In a triangle $AOB, \angle AOB=90°$. If $P$ and $Q$ are the points of trisection of $\overline{AB}$, prove that $O{P}^2+O{Q}^2=\frac{5}{9}A{B}^2$.

Prove by vector method that the measure of the angle between two diagonals of a cube is ${\cos }^{-1}\frac{1}{3}$.