I'm going to assume that the (i, j) thing can be interpreted as coordinates so that I can deal with them as I would complex numbers or plane geometry.
i) Note that any coordinate (x,y) or (i, j), as the case may be, can be described in terms of its distance from the origin (magnitude) and its angle with the x-axis (direction). To do this you can use pythagoras theorem so that the magnitude is the hypotenuse of the right angled triangle formed by the perpendicular from point (i, j) to the x-axis and a line from (i, j) to the origin.
So for A magnitude = √(5<sup>2</sup> + 3<sup>2</sup>) = √(34)
then to find direction you know that:
tanθ=3/5
θ=tan<sup>-1</sup>3/5
= 30'57" or some crap like that
I hope my assumptions have been correct for this 'i' and 'j' thing. If that stuff works then you can use that same method to find B. When you add A +B (ii) then that same method will give you mag. and direc. of them. If you know the angle of A and B from the above method then you can know the angle between them (iv). I'm not going to make assumptions about multiplication though, so i'll leave that to someone else. Teach a man to fish
.
