# 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;