# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#9

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Question

The position vectors of points A and B relative to an origin O are given by

 and

where  is a constant.

i.       In the case where OAB is a straight line, state the value of  and find the unit vector in the direction of .

ii.           In the case where OA is perpendicular to AB, find the possible values of p.

iii.          In the case where , the point C is such that OABC is a parallelogram. Find the position vector of C.

Solution

i.

We are given that;

Since OAB is a straight line, it is evident from the diagram that;

We can see that;

Substituting  in an of the other two equation will yield . We choose;

A unit vector in the direction of  is;

Therefore, for the given case;

We are given that;

We have also found that;

Therefore;

Hence unit vector in the direction of  is;

ii.

If  and  & , then  and  are perpendicular.

Therefore, we need scalar/dot product of  and ;

We are given that;

W need to find .

A vector in the direction of  is;

Therefore, for the given case;

The scalar or dot product of two vectors  and  in component form is given as;

Since ;

For the given case;

Now we have two options;

iii.

We are given that;

In the case where

It is evident from the diagram that;

Since  is a parallelogram, . Therefore we can write;

We can rearrange the equation as;

Therefore, we can write as;