Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#6

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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 mid-point 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 mid-point of AB, therefore;

We have  and , for ;

We are given that;

Hence;

Therefore;

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