Define and use the altitude angle 𝜑.

Calculate the altitude angle \(\varphi\) of \(\overrightarrow{OB}\), in radians correct to two decimal places, where \(O\) is the origin and \(B\) has coordinates \((5, -2, 7)\).

A telecommunications tower is positioned at the origin \(O\) of a three-dimensional coordinate system. A signal is transmitted along the line \(OP\) to a receiver at point \(P\) with coordinates \((5, -7, 11)\), where distances are measured in metres. (a) Define the altitude angle \(\varphi\) of \(OP\) in terms of its relationship to the coordinate planes. (1 mark) (b) Calculate the magnitude of \(\overrightarrow{OP}\). (1 mark) (c) Hence determine the altitude angle \(\varphi\) of \(OP\), in radians, correct to two decimal places. (1 mark)

The altitude angle of \(\overrightarrow{OP}\) is represented as \(\phi\), where \(P\) has coordinates \((5, 8, 12)\). Calculate the altitude angle \(\phi\) in radians.

A drone is positioned at point \(D\) with coordinates \((5, 8, 12)\) relative to a ground-based origin \(O\). (a) Calculate the magnitude of \(\overrightarrow{OD}\). (1 mark) (b) Determine the altitude angle \(\varphi\) of \(\overrightarrow{OD}\), giving your answer in radians correct to two decimal places. (2 marks)

Consider the vector \(\vec{OB}\) with components \((5, -2, 8)\). (a) State the definition of the altitude angle \(\varphi\) for a vector in three-dimensional space. (1 mark) (b) Calculate the altitude angle \(\varphi\) of \(\vec{OB}\) in radians, correct to three decimal places. (1 mark)

An aircraft is tracked at five successive time intervals as it ascends. Position vectors \(\vec{r}\) (in kilometres) relative to an airport control tower and corresponding times \(t\) (in minutes) are recorded in the table below. Determine the altitude angle \(\varphi\) at \(t = 8\) minutes, correct to the nearest degree.