Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2017  MayJun  (P19709/13)  Q#4
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Question
Relative to an origin O, the position vectors of points A, B and C are given by

and 

The point P lies on AB and is such that .
i. Find the position vector of P.
ii. Find the distance OP.
iii. Determine whether OP is perpendicular to AB. Justify your answer.
Solution
i.
We are given that;
We need to find and .
A vector in the direction of is;
Therefore;
ii.
Expression for the length (magnitude) of a vector is;
We have found in (i) that;
Therefore;
iii.
If and & , then and are perpendicular.
Therefore to determine whether OP is perpendicular to AB or not, we need to find out scalar/dot product of and .
The scalar or dot product of two vectors and in component form is given as;


Since ;
For the given case, we have found in (i) that;
and
Since , vectors and are perpendicular.
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