Understand and use the Cartesian form and polar form of a vector.

A vector **u** = 3**i** + 4**j** and a vector **v** has polar form |**v**| = 10 and θ = 210°, where θ is the angle measured anticlockwise from the positive x-axis. Determine the Cartesian form of **v** and hence determine the magnitude of **u** + **v**, correct to two decimal places.

A vector **v** has Cartesian form **v** = 3**i** + 4**j** and a vector **w** has polar form |**w**| = 10, θ = 143°, where θ is measured anticlockwise from the positive x-axis. a) Express **w** in Cartesian form, correct to two decimal places. (1 mark) b) Use the results from part a) to determine |**v** + **w**|, correct to two decimal places. (2 marks)

A navigation system records the positions of three ships relative to a coastal reference point. The table below shows each ship's position in Cartesian form (in kilometres). Convert each position to polar form, giving the magnitude to the nearest 0.1 km and the angle to the nearest degree (measured counterclockwise from the positive x-axis).