Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#3

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Question

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

and

where  is a constant.

     i.       Find the value of  for which angle AOB =.

   ii.       In the case where , find the vector which has magnitude 28 and is in the same direction as .

Solution

     i.
 

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

We are given that;

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

Since ;

Angle AOB= between  and  means  and  are perpendicular to each other.

If  and  & , then  and  are perpendicular.

Therefore, we need to equate scalar/dot product  with ZERO.

   ii.
 

We are given that;

When ;

A vector in the direction of  is;

Therefore to find ;

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

For the given case;

But we are looking for a vector in the direction of  but with magnitude 28. Therefore desired vector in the direction of   will be;

A unit vector in the direction of  is;

For the given case;

Hence desired vector in the direction of  ;

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