Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2012  OctNov  (P19709/13)  Q#9
Hits: 268
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;
Comments