# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/12) | Q#4

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Question

Relative to an origin , the position vectors of three points A and B are given by  and  i.       In the case where , find the unit vector in the direction of .

ii.       Find the values of for which angle .

Solution

i.

We are given that;  In the case where ,  A unit vector in the direction of is;   Therefore for the given case; Therefore, we need to find .

A vector in the direction of is; For the given case;    Hence, unit vector in the direction of (or parallel to) ;   ii.

We recognize that angle AOB is between and . Therefore, we need the dot/scalar product of and .

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 ;    We are given that angle . Therefore,             