Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/12) | Q#7

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Question

The diagram shows a solid cylinder standing on a horizontal circular base with centre O and radius  4 units. Points A, B and C lie on the circumference of the base such that AB is a diameter and angle  BOC = 90o. Points P, Q and R lie on the upper surface of the cylinder vertically above A, B  and C respectively. The height of the cylinder is 12 units. The mid-point of CR is M and N lies on BQ  with BN = 4 units.

Unit vectors  and  are parallel to  and   and unit vector  is vertically upwards.

Evaluate  and hence find angle MPN.

Solution

We are required to evaluate the dot product  first.

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

Since ;

Therefore, we need to find and .

Let us first find .

A vector in the direction of  is;

For the given case;

Therefore, we need the position vectors of points  and  respectively.

First, we find the position vector  of point N.

Let us find the vector .

Since this is the position vector of point , we need coordinates of the point . Consider the diagram below.

·   It is given that is parallel to and we can see that distance of point along  from the origin is 4 units (OA & OB are radii of circle with center O). 

·   It is given that is parallel to and we can see that distance of point along  from the origin is  0 units.

·   It is given that is vertically upwards and we can see that distance of point along  from the origin is 4 units.

Hence, coordinates of .

Now we can represent the position vector of point as follows;

A point has position vector from the origin .
Then the position vector of
is denoted by or .

Next, we find the position vector  of point P.

Let us find the vector . Since this is the position vector of point , we need coordinates of the  point . Consider the diagram below.

·   It is given that is parallel to and we can see that distance of point along  from the origin is  -4 units. 

·    It is given that is parallel to and we can see that distance of point along  from the origin is  0 units.

·   It is given that is vertically upwards and we can see that distance of point along  from the origin is 12 units.

Hence, coordinates of .

Now we can represent the position vector of point as follows;

A point has position vector from the origin . Then the position vector of is denoted by or .

Now we can find .

Let us first find .

A vector in the direction of  is;

For the given case;

Therefore, we need the position vectors of points  and  respectively.

First, we find the position vector  of point M.

Let us find the vector .

Since this is the position vector of point , we need coordinates of the point . Consider the  diagram below.

·   It is given that is parallel to and we can see that distance of point along  from the origin is  0 units. 

·   It is given that is parallel to and we can see that distance of point along  from the origin is  4 units (OA & OB & OC are radii of circle with center O).

·   It is given that is vertically upwards and we can see that distance of point along  from the  origin is 6 units (M is mid-point of CR which is 12 units).

Hence, coordinates of .

Now we can represent the position vector of point as follows;

A point has position vector from the origin . Then the position vector of is denoted by or .

We have already found position vector  of point P.

Now we can find .

Finally, we are able to find dot product .

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

Since ;

Now we need to find the angle MPN.

We recognize that  is angle between and  .
Hence, we use
scalar/dot product of and to find angle .

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

Since ;

We have from (i) that;

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;

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