2022 RI J1 CT Q8

The earth can be modelled as a sphere, centered at the origin \(O\) with radius \(r\).

A point on the sphere with longitude \(\varphi \) and latitude \(\theta \), both measured in degrees, has position vector \(\overrightarrow{OP}=\left( \begin{matrix} r\sin \theta \cos \varphi \\ r\sin \theta \sin \varphi \\ r\cos \theta \\ \end{matrix} \right)\).

The following points on the sphere have the corresponding longitudes and latitudes, measured in degrees:

1. Show that the cartesian equation of the plane containing \(O\), \(F\) and \(R\) is given by \(\sqrt{3}y=x\).[3]

A spherical angle is the angle between two arcs on a sphere, and is measured by the acute angle between the planes containing the arcs and the centre of the sphere.

2. Find the spherical angle between the arcs \(FR\) and \(IC\).[4]

An aircraft travels from \(F\) to \(C\) along the arc \(FC\). Let \(\alpha \) be the angle between the vectors \(\overrightarrow{OF}\) and \(\overrightarrow{OC}\).

3. Show that \(\overrightarrow{OF}\cdot \overrightarrow{OC}={{r}^{2}}\cos \alpha \).[1]
4. Hence find the length of the arc \(FC\) to the nearest km, given that the radius of earth can be taken to be \(6378\)km.[3]
5. Give a reason why the actual distance travelled by the aircraft is usually greater than the arc length \(FC\) calculated in part (iv).[1]