# 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;     