Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2018  FebMar  (P19709/12)  Q#7
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Question
Fig. 1 shows a rectangle with sides of 7 units and 3 units from which a triangular corner has been removed, leaving a 5sided polygon OABCD. The sides OA, AB, BC and DO have lengths of 7 units, 3 units, 3 units and 2 units respectively. Fig. 2 shows the polygon OABCD forming the horizontal base of a pyramid in which the point E is 8 units vertically above D. Unit vectors , and are parallel to , and respectively.
i. Find and the length of CE.
ii. Use a scalar product to find angle ECA, giving your answer in the form where m and n are integers.
Solution
i.
We are required to find .
A vector in the direction of
For the given case;
Therefore, we need the position vectors
First we find the position vector
Let us find the vector
· 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
Next we find the position vector
Let us find the vector
· 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
Now we can find
Next we are required to find the length of
Expression for the length (magnitude) of a vector is;
ii.
We recognize that
Hence we use scalar/dot product of
The scalar or dot product of two vectors


Since
Therefore, we need to find
We have from (i) that;
Next, let us find the vector
A vector in the direction of
For the given case;
Therefore, we need the position vectors
From (i) we already have
First we find the position vector
Since this is the position vector of point
· 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
Now we can find
Now we find the scalar product of
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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