# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | Oct-Nov | (P1-9709/01) | Q#4

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Question

The position vectors of points and are and respectively, relative to an origin .

i.
Calculate angle .

ii.       The point is such that . Find the unit vector in the direction of .

Solution

i.

We recognize that is angle between and .
Hence we use
scalar/dot product of and .  The scalar or dot product of two vectors and in component form is given as;     Since ; Therefore for the given case;   The scalar or dot product of two vectors and is number or scalar , where is the angle between the directions of and . Where    For the given case;     Therefore;  Equating both scalar/dot products we get;      Hence the angle AOB is .

ii.

We are given that First we need to find .

A vector in the direction of is;   Therefore, for the given case;  Since;   A vector in the direction of is; Therefore;     A unit vector in the direction of is;   For the given case;   