# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#5

Question

Relative to an origin O, the position vectors of the points A and B are given by

 and

i.
For the case where OA is perpendicular to OB, find the value of p.

ii.       For the case where OAB is a straight line, find the vectors  and . Find also the length of            the line OA.

Solution

i.

We recognize that angle AOB is between  and .

We are given that AOB is a right angle.

If  and  & , then  and  are perpendicular.

Therefore;

Now, we need the scalar/dot product of  and .

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

Since ;

ii.

If three points A,B and C lie on the same straight line then two vectors starting from the the same  point, say A, but ending at both other different points, B or C,  must be parallel.

Conversely if  then all three points A, B and C lie sin the same straight line.

We are given;

 and

We are also given that OAB is a straight line, therefore, .

Lets first find .

A vector in the direction of  is;

For the given case;

Therefore;

Two vectors A and B are parallel if and only if they are scalar multiples of one another. If ,  then;

where k is a constant not equal to zero.

Therefore, for the given case;

It is evident that if two vectors are scalar multiples then;

Therefore;

Substituting ;

Therefore, we can write vectors  and  as;

Expression for the length (magnitude) of a vector is;

Magnitude of .