# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/11) | 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;     