Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2018  OctNov  (P19709/13)  Q#6
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Question
The diagram shows a solid figure OABCDEFG with a horizontal rectangular base OABC in which OA = 8 units and AB = 6 units. The rectangle DEFG lies in a horizontal plane and is such that D is 7 units vertically above O and DE is parallel to OA. The sides DE and DG have lengths 4 units and 2 units respectively.
Unit vectors and and are parallel to and and respectively.
Use a scalar product to find angle OBF, giving your answer in the form , where and are integers.
Solution
We are required to find the angle OBF.
We recognize that is angle between
Hence, we use scalar/dot product of
The scalar or dot product of two vectors


Since
Therefore, need to find
First, we find
Since this is the position vector of point
Consider the diagram below.
· It is given that
· It is given that
· It is given that
Hence, coordinates of
Now we can represent the position vector of point
A point
denoted by
Next, we find
A vector in the direction of
For the given case;
Therefore, we need the position vectors
Therefore, we need to find
Since this is the position vector of point
Consider the diagram below.
· It is given that
· It is given that
· It is given that
Hence, coordinates of
Now we can represent the position vector of point
A point
We have already found position vector
Now we can find
Finally, we are able to find dot product
The scalar or dot product of two vectors


Since
The scalar or dot product of two vectors
Where





For the given case;
Therefore;
Equating both scalar/dot products we get;
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