# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/12) | Q#5

Hits: 205

Question

The diagram shows a three-dimensional shape. The base OAB is a horizontal  triangle in which angle AOB is 90o. The side OBCD is a rectangle and the side OAD  lies in a vertical plane. Unit vectors  and are parallel to OA and OB respectively  and the unit vector  is vertical.
The position vectors of A, B and D are given by
,  and

i.       Express each of the vectors  and in terms of ,  and .

ii.       Use a scalar product to find angle CAD.

Solution

i.

We are required to workout .

A vector in the direction of  is;

For the given case;

Therefore, we need the position vectors of points  and .

We are  given that position vectors of A, B and D are given as;

Therefore;

Next we are required to find .

A vector in the direction of  is;

For the given case;

Therefore, we need the position vectors of points  and .

Next we find .

To find consider the diagram below.

It is evident from the diagram that;

We are given that OBCD is a rectangle, therefore, its opposite sides are parallel and equal. Hence,  and are equal and parallel and we can write above equation as;

We are given that;

Hence;

Now we can find ;

ii.

We are required to find the angle CAD.

It is evident from the diagram that angle CAD is between  and .

Therefore, we use scalar/dot product of and to find angle CAD.

From (i) we have;

It can be written as;

We need to find .

To find consider the diagram below.

It is evident from the diagram that;

Hence;

We are given that;

Therefore;

It can be written as;

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

For the given case;

Therefore;

Equating both scalar/dot products we get;

Hence the angle CAD is .