# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#8

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Question

The points , ,  and  have position vectors , ,   and  respectively.

i.
Use a scalar product to show that  and  are perpendicular.

ii.       Show that  and  are parallel and find the ratio of the length of  to the length of .

Solution

We have the position vectors for points , ,  and ;

i.

First of all we find  and .

For  .

For  .

Now we find the scalar/dot product of  and .

The scalar or dot product of two vectors  and  in component form is given as;

Since ;

If  and  & , then  and  are perpendicular.

Hence,  and   are perpendicular.

ii.

First of all we find   and  .

From (i) we have;

For  .

Now we find the scalar/dot product of  and .

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;

Now we find out  and .

Now we can substitute the values in the equation;

Hence  and  are parallel.

Now we find the ratio of the length of  to the length of .