# 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;   