# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/12) | Q#9

Question

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

 and

i.
Use a scalar product to find angle AOB.

ii.
Find the vector which is in the same direction as  and of magnitude 15 units.

iii.
Find the value of the constant p for which  perpendicular to .

Solution

i.

It is evident that angle AOB is between  and .

We are given that;

Next, we need scalar/dot product of  and .

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

Since ;

For the given case;

Scalar/Dot product is also defined as below.

The scalar or dot product of two vectors  and  is number or scalar , where  is the  angle between the directions of  and

For;

Therefore, we need to find  and .

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

Therefore;

Hence;

Equating both scalar/dot products found above;

Therefore;

ii.

We are required to find a vector whose magnitude is given but not the direction vector.

However, we are given that desired vector is in the same direction as  .

Therefore, we need the direction vector of  .

A unit vector in the direction of  is;

For the given case;

It is evident that first we need to find  and then .

A vector in the direction of  is;

For the given case;

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

Therefore;

Hence;

For a vector in the direction of  with magnitude of 15;

iii.

If  and  & , then  and  are perpendicular.

Therefore we need to find the scalar/dot product of   and  and equate it with ZERO.

We are given that;

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

Since ;

For the given case;