Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/12) | Q#8

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


where p and q are constants.

     i.       In the case where , use a scalar product to find angle AOB.

   ii.       In the case where  is parallel to , find the values of p and q.



It is evident that angle AOB is between  and .

We are given that;

As we are given ;

Next, 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;

Scalar/Dot product is also defined as below.

The scalar or dot product of two vectors  and  is number or scalar , where  is the  angle between the directions of  and  .


Therefore, we need to find  and .

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



Equating both scalar/dot products found above;



We are given that  is parallel to .

Two vectors are parallel when they are scalar multiples of each other. In other words, if you can  multiply one vector by a constant and end up with the other vector.

We are given that;

Therefore, if  is parallel to ;

Now we need to find .

A vector in the direction of  is;

For the given case;

We are given that;


Now we can write expression which represents that  is parallel to ;

We can compare all the three terms of the two vectors.

We can see that from these three equations, we can have;

Therefore, we substitute  in  to find values of p.

Now we can substitute  and  in  to find values of q;