Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2005  OctNov  (P19709/01)  Q#4
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Question
Relative to an origin , the position vectors of the points , and are given by


where is a constant.
i. In the case where , use a scalar product to show that .
ii. Find the values of q for which the length of is 6 units.
Solution
i.
We are given that . Therefore,


The angle POQ is the angle between and . We use scalar/dot product of and to find .
The scalar or dot product of two vectors and in component form is given as;


Since ;
Therefore, for the given case;
The scalar or dot product of two vectors and is number or scalar , where is the angle between the directions of and .
Where




Therefore for the given case;
Therefore for the given case;
Hence the scalar/ dot product;
Equating both scalar/dot products;
ii.
We are given that;
A vector in the direction of is;
Therefore, for the given case;
Expression for the length (magnitude) of a vector is;
Therefore;
Since, we are given that;
Now we have two options;






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