# 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

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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;