Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2010  MayJun  (P19709/13)  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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