Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | Oct-Nov | (P1-9709/01) | Q#4

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Question

The position vectors of points  and  are  and   respectively, relative to an origin .


i.       
Calculate angle .

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

Solution


i.
 

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

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

Since ;

Therefore for the given case;

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

Where

For the given case;

Therefore;

Equating both scalar/dot products we get;

Hence the angle AOB is .


ii.
 

We are given that

First we need to find .

A vector in the direction of  is;

Therefore, for the given case;

Since;

A vector in the direction of  is;

Therefore;

A unit vector in the direction of  is;

For the given case;

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