Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2019  MayJun  (P19709/12)  Q#8
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Question
The position vectors of points A and B, relative to an origin O, are given by

and 

Where is a constant.
i.Find the value of for which the angle AOB=90^{o}.
ii.Find the values of for which the lengths of OA and OB are equal.
The point C is such that .
iii.In the case where , find the unit vector in the direction of .
Solution
i.
It is evident that angle AOB is between and .
We are given that angle AOB is .
If and & , then and are perpendicular.
Therefore, we need scalar/dot product of and ;
We are given that;

and 

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


Since ;
For the given case;
ii.
We are required to find the value of for which;
Expression for the length (magnitude) of a vector is;
Therefore;








We are given that;
iii.
We are required to find unit vector in the direction of .
We are given that and;

and 

Therefore;

and 

We are also given that;
A vector in the direction of is;
Therefore;
We are required to find unit vector in the direction of .
A unit vector in the direction of is;
Therefore, for the given case;
Therefore, we need vector and its magnitude .
We already have vector and need to find its magnitude .
Expression for the length (magnitude) of a vector is;
Therefore;
Now we can find unit vector in the direction of .
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