# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/11) | Q#10

Question

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

 and

respectively, where k is a constant.

i.
Find the value of k in the case where angle AOB=90o.

ii.
Find the possible values of k for which the lengths of AB and OC are equal.

The point D is such that  is in the same direction as  and has magnitude 9 units. The point E  is such that  is in the same direction as  and has magnitude 14 units.

iii.
Find the magnitude of  in the form  where n is an integer.

Solution

i.

It is evident that angle AOB is between  and .

We are given that angle AOB is .

If  and  & , then  and  are perpendicular.

Therefore, we need scalar/dot product of  and ;

We are given that;

 and

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

Since ;

For the given case;

ii.

We are given that;

We are given;

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

Next we need to find .

A vector in the direction of  is;

We are given that;

 and

Therefore, for the given case;

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

Therefore, according to the given condition;

Now we have two options;

iii.

We are required to find the magnitude of .

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

Therefore, first we need to find .

A vector in the direction of  is;

Hence;

Now we first find .

We are given that  and is in the same direction as  .

A unit vector in the direction of  is;

Conversely, a vector  in a given direction is;

Therefore;

We are given that  is in the same direction as . Therefore;

Let’s find unit vector of ;

A unit vector in the direction of  is;

We are given;

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

Therefore;

Hence;

Let’s now find .

We are given that  and is in the same direction as  .

A unit vector in the direction of  is;

Conversely, a vector  in a given direction is;

Therefore;

We are given that  is in the same direction as . Therefore;

Let’s find unit vector of ;

A unit vector in the direction of  is;

We are given;

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

Therefore;

Hence;

Finally, we can write;

Now we can find magnitude of .

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