Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2011  OctNov  (P19709/13)  Q#6
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Question
Relative to an origin , the position vectors of points A and B are and , respectively.
i. Use a scalar product to calculate angle BOA.
The point C is the midpoint of AB. The point D is such that .
ii. Find .
Solution
i.
We recognize that angle BOA is between and
We are given that;
The scalar or dot product of two vectors


Since
The scalar or dot product of two vectors
Where




Therefore, for the given case;
Therefore;
Hence;
Now we can equate the two equations of
Hence the angle BOA is
ii.
A vector in the direction of
For the given case;
Consider the diagram given below.
It is evident from the diagram that;
We are given that point D is such that
It is also evident from the diagram that;
Hence;
We are also given that point C is the midpoint of AB, therefore;
We have
We are given that;
Hence;
Therefore;
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