Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2016  OctNov  (P19709/12)  Q#9
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Question
Relative to an origin O, the position vectors of points A, B and C are given by


and 

i. Use a scalar product to find angle AOB.
ii. Find the vector which is in the same direction as and of magnitude 15 units.
iii. Find the value of the constant p for which perpendicular to .
Solution
i.
It is evident that angle AOB is between and .
We are given that;
Next, we need 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;
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;
ii.
We are required to find a vector whose magnitude is given but not the direction vector.
However, we are given that desired vector is in the same direction as .
Therefore, we need the direction vector of .
A unit vector in the direction of is;
For the given case;
It is evident that first we need to find and then .
A vector in the direction of is;
For the given case;
Expression for the length (magnitude) of a vector is;
Therefore;
Hence;
For a vector in the direction of with magnitude of 15;
iii.
If and & , then and are perpendicular.
Therefore we need to find the scalar/dot product of and and equate it with ZERO.
We are given that;
The scalar or dot product of two vectors and in component form is given as;


Since ;
For the given case;
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