Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#6

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Question

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


i.       
Use a scalar product to find angle ABC.

   ii.       Find the perimeter of triangle ABC, giving your answer correct to 2 decimal places.

Solution


i.
 

We recognize that angle ABC is between vectors  and  Therefore, first we find vectors  and .

A vector in the direction of  is;

For the given case;

Now we need to find the scalar/dot product of   and . We have;

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

Since ;

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

Therefore for the given case;

Therefore for the given case;

Hence the scalar/ dot product;

Equating both scalar/dot products;

Hence the angle ABC is .


ii.
 

We recognize that perimeter of the triangle will be;

From (i) we have;

Now we need . This can be found as follows.

We are given that;

A vector in the direction of  is;

For the given case;

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

Therefore;

From (i) we have;

Therefore;

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