Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2016  MayJun  (P19709/13)  Q#9
Question
The position vectors of points A, B and C relative to an origin O are given by


and 

where p is a constant.
i. Find the value of p for which the lengths of AB and CB are equal.
ii. For the case where p=1, use a scalar product to find angle ABC.
Solution
i.
We are given that;
First we need to find .
A vector in the direction of is;
We are given that;

and 

Therefore, for the given case;
Expression for the length (magnitude) of a vector is;
Next we need to find .
A vector in the direction of is;
We are given that;

and 

Therefore, for the given case;
Expression for the length (magnitude) of a vector is;
Therefore, according to the given condition;
ii.
It is evident that angle ABC is between and . It is also quite fine to visualize the angle ABC between and .
From (i) we have found that;
Since p=1;
Next, we need scalar/dot product of and .
The scalar or dot product of two vectors and in component form is given as;


Since ;
For the given case;
Scalar/Dot product is also defined as below.
The scalar or dot product of two vectors and is number or scalar , where is the angle between the directions of and .
For ;
We have found from (i) that;
Since p=1;
Equating both scalar/dot products found above;
Therefore;
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