Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | May-Jun | (P1-9709/01) | Q#6

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Question

Relative to an origin , the position vectors of points  and  are given by


i.      
Find the value of  and hence state whether angle AOB is acute, obtuse or a right angle.

   ii.       The point  is such that . Find the unit vector in the direction of OX.

Solution


i.
 

We recognize that  is angle between  and  .
Hence we use
scalar/dot product of  and . We have both    and .

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

Since ;

For the given case;

If  and  & , then  and  are perpendicular.

If , then angle between   and  is acute.

If , then angle between  and  is obtuse.

Therefore angle AOB is obtuse.


ii.
 

From the given condition;

We need  to substitute in above equation to find .

A vector in the direction of  is;

Therefore, for the given case;

Hence, we can write;

A unit vector in the direction of  is;

Now, unit vector in the direction of  can be found as follows.

 Hence, we need . This can be found from  as follows.

We can find  as follows;

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