# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/13) | Q#7

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Question

Three points, O, A and B, are such that  and  , where is a constant.

i. Find the values of  for which  is perpendicular to .

ii. The magnitudes of   and   are  and  respectively. Find the value of  for which .

iii. Find the unit vector in the direction of  when .

Solution

i.

We are given that;

If  and  & , then  and  are perpendicular.

Therefore, if  and  are perpendicular then

We next find the scalar/dot product of  and  and equate that equal to zero.

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

Since ;

Since we are given  .

Now we have two options.

ii.

Expression for the length (magnitude) of a vector is;

Therefore we find first the magnitudes of  and .

 We are given that We are also given that . Therefore;

Hence;

iii.

We are given that;

Therefore;

A unit vector in the direction of  is;

Therefore, first we need to find .

A vector in the direction of  is;

For the ;

We are given above two vectors. Hence;

Next we need magnitude of .

Hence;