Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#4

Hits: 930

Question

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

and


i.       
Find the cosine of angle AOB.

The position vector of C is given by .

 

   ii.       Given that AB and OC have the same length, find the possible values of k.

Solution

     i.
 

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

Now we need to find .

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

Since ;

We are given;

Therefore;

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

where

Therefore, for the given case;

Therefore;

Hence;

Now we can equate the two equations of ;

   ii.
 

We are given that;

First we need to find .

A vector in the direction of  is;

For the given case;

We are given;

Therefore;

We are given that  and  are of same length, that means;

Magnitude of a vector ;

Therefore;

Evidently, it is a quadratic equation for which solution is;

For the above equation;

Therefore;

Now we have two options.

Comments