Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (You may assume that the trapezium is not a parallelogram.)

For any two vectors $$u$$ and $$v,$$ prove that
(a) $${\left( {u\,.\,v} \right)^2} + {\left| {u \times v} \right|^2} = {\left| u \right|^2}{\left| v \right|^2}$$ and
(b) $$\left( {1 + {{\left| u \right|}^2}} \right)\left( {1 + {{\left| v \right|}^2}} \right) = {\left( {1 - u.v} \right)^2} + {\left| {u + v + \left( {u \times v} \right)} \right|^2}.$$

The SI unit of inductance, the henry can be written as

Let [$${\mathrm\varepsilon}_\mathrm o$$] denote the dimentional formula of the permittivity of the vacuum, and [$$\mu_o$$] that of the permeability of the vacuum.If M = mass, L = length, T = time and I = electric current,