Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2017  MayJun  (P19709/12)  Q#8
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Question
Relative to an origin , the position vectors of points A and B are given by


and 

where p and q are constants.
i. In the case where , use a scalar product to find angle AOB.
ii. In the case where is parallel to , find the values of p and q.
Solution
i.
It is evident that angle AOB is between and
We are given that;


As we are given




Next, we need scalar/dot product of
The scalar or dot product of two vectors


Since
For the given case;
Scalar/Dot product is also defined as below.
The scalar or dot product of two vectors
For
Therefore, we need to find
Expression for the length (magnitude) of a vector is;
Therefore;







Hence;
Equating both scalar/dot products found above;
Therefore;
ii.
We are given that
Two vectors are parallel when they are scalar multiples of each other. In other words, if you can multiply one vector by a constant and end up with the other vector.
We are given that;
Therefore, if
Now we need to find
A vector in the direction of
For the given case;
We are given that;


Therefore;
Now we can write expression which represents that
We can compare all the three terms of the two vectors.



We can see that from these three equations, we can have;
Therefore, we substitute
Now we can substitute
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