(1) If a+ib is the multiplicative inverse then (4-3i)(a+ib)=1.
Expanding: 4a+3b+i(4b-3a)=1, so 4a+3b=1 and 4b-3a=0, so b=3a/4.
4a+9a/4=1, 25a/4=1 and a=4/25. Therefore b=3/25.
The inverse is therefore (4+3i)/25.
(2) What is the meaning of "top point" of side BC?
Vector AB, as an example, means the direction is from A to B. This is the way the vectors below are intended to be read. The first letter is the "from" and the second letter is the "to".
Vectors AB+AC=BC; vectors AB+AD=BD=(2/3)BC, 3AB+3AD=2BC, so 3AD=2BC-3AB. If 2AB+BC=2BC-3AB, this implies 5AB=BC, which is not true, because |AB|=|BC|=a, the side length.
AD•BC=|AD||BC|cos(20), because angle A is trisected so DAC=20 degrees. BC^2=a^2 where a is the side length. But |AD||BC|cos(20)>0 and a^2>0, making the right side positive always, while -1/6 makes the right side negative.
It appears that neither (a) nor (b) is true.
(3) |AO|=2|OD| and |AD|=3|OD| because the medians intersect at a third of their lengths and |AD|=|AO|+|OD|.
Vector AD is the average of vectors u and v, which is their sum divided by 2, but the magnitude of AO is 2/3 the magnitude of AD, so vector AO, having the same direction as AD,=(2/3)(u+v)/2=(1/3)(u+v).