Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2010  MayJun  (P19709/11)  Q#10
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Question
The diagram shows the parallelogram OABC. Given that and , find
i. the unit vector in the direction of .
ii. the acute angle between the diagonals of the parallelogram,
iii. the perimeter of the parallelogram, correct to 1 decimal place.
Solution
i.
A unit vector in the direction of is;
Therefore, for the given case;
It is evident from the diagram above that;
Since , we can write;
We are given both and ;
Therefore;
Now, unit vector in the direction of can be found as follows.
We can find as follows;
ii.
We recognize that diagonals of parallelogram are and . It is evident from the given diagram that;
We are given both and ;
Therefore;
From (i), we have;
To find the angle between and we use the scalar/dot product of these two vectors.
The scalar or dot product of two vectors and in component form is given as;


Since ;
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;
Hence;
Now we can equate the two equations of ;
Since we are required to find acute angle;
Hence the angle acute angle between the diagonals of the given parallelogram is .
iii.
It is evident that perimeter of the parallelogram is;
Since;
We can rewrite the perimeter as;
We are given both and ;
Expression for the length (magnitude) of a vector is;
Therefore;
Hence;
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