Find the angle between the two vectors  and  , giving your answer in radians, correct to 1 decimal place.   (2 marks)

Find the cosine of the acute angle between the vectors    and  .   (2 marks)

Find the angle between the vectors

,

giving your answer to the nearest degree.  (3 marks)

Two planes are flying in directions given by the vectors    and  .

A person in the flight control centre is plotting their paths on a map.

Calculate the acute angle between their projected flight paths, giving your answer correct to one decimal place.  (2 marks)

Find the angle between the vectors    and  , giving the angle in degrees correct to 1 decimal place. (3 marks)

A triangle is formed in three-dimensional space with vertices  ,   and  .

Find the size of , giving your answer to the nearest degree.  (3 marks)

Find the angle between the vectors    and  , giving the angle in degrees correct to 1 decimal place. (3 marks)

Let    and  , where the acute angle between these vectors is  .

The value of    is

The acute angle    is the angle between the vectors    and  .

Find the exact value of  .   (2 marks)

Relative to a fixed origin, the points , and are defined respectively by the position vectors    and    where is a real constant.

Given that the magnitude of angle is  , find .  (4 marks)