Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2015  OctNov  (P19709/12)  Q#7
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Question
Relative to an origin O, the position vectors of the points A, B and C are given by


and 

i. In the case where ABC is a straight line, find the values of p and q.
ii. In the case where angle BAC is 90^{o}, express q in terms of p.
iii. In the case where p=3 and the lengths of AB and AC are equal, find the possible values of q.
Solution
i.
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 ABC is a straight line, therefore, .
Lets first find and .
A vector in the direction of is;
For the given case;
Therefore;
Next we need to 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 ;
We also have;
Substituting ;
ii.
We are given that angle BAC is 90^{o}.
If and & , then and are perpendicular.
Therefore, and are perpendicular.
Hence .
The scalar or dot product of two vectors and in component form is given as;


Since ;
We are given that;
Hence;
iii.
We have already found and in (i)
We are given that , therefore;
We are also given that the lengths of AB and AC are equal. Therefore;
Lets now find the lengths (magnitudes) of both and and equate them.
Expression for the length (magnitude) of a vector is;
Hence;








Since ;
Now we have two options;




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