Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2018  MayJun  (P19709/11)  Q#7
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Question
Relative to an origin O, the position vectors of points A, B and C are given by


and 

i. Find .
ii. The point M is the midpoint of AC. Find the unit vector in the direction of .
iii. Evaluate and hence find angle BAC.
Solution
i.
We are required to find the vector .
A vector in the direction of is;
For the given case;
ii.
We are required to find the unit vector in the direction of .
A unit vector in the direction of is;
For the given case;
It is evident that first we need to find and then .
We are also given that point M is the midpoint of the AC. Therefore;
Next we need .
Expression for the length (magnitude) of a vector is;
Therefore;
Hence;
iii.
We are required to find and then find angle BAC.
The scalar or dot product of two vectors and in component form is given as;


Since ;
Therefore, to find we need vectors and .
We have already found in (i).
Therefore, we need to find first.
A vector in the direction of is;
For the given case;
Now we can find .
It is evident that angle BAC is between and .
We have found above that;
Scalar/Dot product is also defined as below.
The scalar or dot product of two vectors and is number or scalar , where is the angle between the directions of and .
For;
Therefore, we need to find and .
Expression for the length (magnitude) of a vector is;
Therefore;








Hence;
Equating both scalar/dot products found above;
Therefore;
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