Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#4

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Question

Relative to an origin O, the position vectors of points A, B and C are given by

and

The point P lies on AB and is such that .


i.       
Find the position vector of P.


ii.       
Find the distance OP.


iii.       
Determine whether OP is perpendicular to AB. Justify your answer.

Solution

     i.
 

We are given that;

We need to find  and .

A vector in the direction of  is;

Therefore;


ii.
 

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

We have found in (i) that;

Therefore;

  iii.
 

If  and  & , then  and  are perpendicular.

Therefore to determine whether OP is perpendicular to AB or not, we need to find out 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, we have found in (i) that;

    and   

Since , vectors  and  are perpendicular.

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