# 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; 