# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#10

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Question

The diagram shows triangle , in which the position vectors of  and  with respect to  are given by

 &

is a point on  such that , where  is a
constant.

i.       Find angle .

ii.     Find  in terms of  and vectors ,  and .

iii.   Find the value of  given that  is perpendicular to .

Solution

i.

We recognize that angle AOB is between  and . Therefore, we use the scalar/dot product of these two to find angle AOB.

The scalar or dot product of two vectors  and  in component form is given as;

Since ;

We have;

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 ;

Hence the angle AOB is .

ii.

A vector in the direction of  is;

Therefore;

Now we need  and . We already have;

To find  we utilize the given condition;

We are given that;

Therefore;

Hence;

iii.

If  and  & , then  and  are perpendicular.

Therefore, we need scalar/dot product of  and ;

The scalar or dot product of two vectors  and  in component form is given as;

Since ;

For the given case;