# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | Oct-Nov | (P1-9709/01) | Q#4

Hits: 81

Question

Relative to an origin , the position vectors of the points , and are given by  where is a constant.

i.
In the case where , use a scalar product to show that .

ii.       Find the values of q for which the length of is 6 units.

Solution

i.

We are given that . Therefore,  The angle POQ is the angle between and . We use scalar/dot product of and to find .

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    Therefore for the given case; Therefore for the given case;    Hence the scalar/ dot product;  Equating both scalar/dot products;    ii.

We are given that; A vector in the direction of is; Therefore, for the given case; Expression for the length (magnitude) of a vector is;  Therefore; Since, we are given that;        Now we have two options;      