Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2018  OctNov  (P19709/11)  Q#8
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Question
The diagram shows a solid figure OABCDEF having a horizontal rectangular base OABC with OA = 6 units and AB = 3 units. The vertical edges OF, AD and BE have lengths 6 units, 4 units and 4 units respectively. Unit vectors , and are parallel to , and respectively.
i. Find .
ii. Find the unit vector in the direction of .
iii. Use a scalar product to find angle EFD.
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
denoted by
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
ii.
We are required to find unit vector in the direction of
A unit vector in the direction of
Therefore, for the given case;
Therefore, we need vector
Let’s first find
A vector in the direction of
Therefore, for the given case;
We have found in (i) that;
We need to find
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
denoted by
Hence;
Now we find magnitude of
Expression for the length (magnitude) of a vector is;
Therefore;
Now we can find unit vector in the direction of
iii.
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;
We have from (ii) that;
Now we find the scalar product of
The scalar or dot product of two vectors
the angle between the directions of
Where





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