Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2015  MayJun  (P19709/12)  Q#9
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Question
Relative to an origin , the position vectors of points A and B are given by

and 

i. Use a vector method to find angle AOB.
The point C is such that .
ii. Find the unit vector in the direction of
iii. Show that triangle OAC is isosceles.
Solution
i.
We recognize that angle AOB is between and
We are given that;
The scalar or dot product of two vectors


Since
The scalar or dot product of two vectors
where
Therefore, for the given case;
Therefore;
Hence;
Now we can equate the two equations of
ii.
A unit vector in the direction of
Therefore for the given case;
Therefore, we need to find
We are given that;
A vector in the direction of
We are given that;
Therefore;
Hence, unit vector in the direction of (or parallel to)
iii.
An isosceles triangle is a triangle with two equal sides.
We are to show that OAC is an isosceles triangle. If OAC is an isosceles triangle then either A=OC or AC=OC.
Let us try to show OA=OC.
We are given that;
We have found in (ii);
We can find magnitudes of both vectors.
Expression for the length (magnitude) of a vector is;
Hence OAC is an isosceles triangle.
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