Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2015  OctNov  (P19709/13)  Q#5
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Question
Relative to an origin O, the position vectors of the points A and B are given by

and 

i. For the case where OA is perpendicular to OB, find the value of p.
ii. For the case where OAB is a straight line, find the vectors and . Find also the length of the line OA.
Solution
i.
We recognize that angle AOB is between and .
We are given that AOB is a right angle.
If and & , then and are perpendicular.
Therefore;
Now, we need the scalar/dot product of and .
The scalar or dot product of two vectors and in component form is given as;


Since ;
ii.
If three points A,B and C lie on the same straight line then two vectors starting from the the same point, say A, but ending at both other different points, B or C, must be parallel. .
Conversely if then all three points A, B and C lie sin the same straight line.
We are given;

and 


We are also given that OAB is a straight line, therefore, .
Lets first find .
A vector in the direction of is;
For the given case;
Therefore;
Two vectors A and B are parallel if and only if they are scalar multiples of one another. If , then;
where k is a constant not equal to zero.
Therefore, for the given case;
It is evident that if two vectors are scalar multiples then;



Therefore;
Substituting ;
Therefore, we can write vectors and as;
Expression for the length (magnitude) of a vector is;
Magnitude of .
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