Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2011  OctNov  (P19709/13)  Q#6
Hits: 402
Question
Relative to an origin , the position vectors of points A and B are and , respectively.
i. Use a scalar product to calculate angle BOA.
The point C is the midpoint of AB. The point D is such that .
ii. Find .
Solution
i.
We recognize that angle BOA is between and . Therefore, we need the scalar/dot product of these two to find angle BOA.
We are given that;
The scalar or dot product of two vectors and in component form is given as;


Since ;
The scalar or dot product of two vectors and is number or scalar , where is the angle between the directions of and .
Where




Therefore, for the given case;
Therefore;
Hence;
Now we can equate the two equations of ;
Hence the angle BOA is .
ii.
A vector in the direction of is;
For the given case;
Consider the diagram given below.
It is evident from the diagram that;
We are given that point D is such that . Therefore;
It is also evident from the diagram that;
Hence;
We are also given that point C is the midpoint of AB, therefore;
We have and , for ;
We are given that;
Hence;
Therefore;
Comments