Question

Relative to the origin O, two points A and B have position vectors a and b respectively. A line, l, passes through A and is parallel to b. It is given that b is a unit vector.

(i) Write down a vector equation of l. Show that the position vector of the point N on l such that the length |ON| is the shortest is given by a-(a.b)b.

(ii) The point M is on AN produced such that kAN = NM, where k is a constant. Given that the position of M is a-5(a.b)b, find k.

It is given that |a|=2 and a.b=1/3.

(iii) Give the geometric meaning of |b x (a-b)| and find its exact value.

(iv) C is a point such that OC bisects the angle AOB. Write down, in terms of a and b, a possible position vector of C.

Any help would be greatly appreciated! Thank you :)