Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2011  OctNov  (P19709/12)  Q#3
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Question
Relative to an origin , the position vectors of points A and B are given by

and 

where is a constant.
i. Find the value of for which angle AOB =.
ii. In the case where , find the vector which has magnitude 28 and is in the same direction as .
Solution
i.
We recognize that angle AOB is between and . Therefore, we need the scalar/dot product of these two.
We are given that;
The scalar or dot product of two vectors and in component form is given as;


Since ;
Angle AOB= between and means and are perpendicular to each other.
If and & , then and are perpendicular.
Therefore, we need to equate scalar/dot product with ZERO.
ii.
We are given that;
When ;
A vector in the direction of is;
Therefore to find ;
Expression for the length (magnitude) of a vector is;
For the given case;
But we are looking for a vector in the direction of but with magnitude 28. Therefore desired vector in the direction of will be;
A unit vector in the direction of is;
For the given case;
Hence desired vector in the direction of ;
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