Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2011  MayJun  (P19709/13)  Q#5
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Question
In the diagram, OABCDEFG is a rectangular block in which cm and cm. Unit vectors , and are parallel to , and respectively. The point P is the midpoint of DG, Q is the centre of the square face CBFG and R lies on AB such that AR = 4 cm.
i. Express each of the vectors and in terms of , and .
ii. Use a scalar product to calculate angle RPQ.
Solution
i.
We are required to find vectors and
First let’s find vector .
A vector in the direction of is;
For the given case;
Therefore, we need the position vectors of points and .
A point has position vector from the origin . Then the position vector of is denoted by or .
Hence we need coordinates of the points and .
First, let’s find coordinates of point . Consider the diagram.
It is given that is parallel to and it is evident from the diagram that is equal and parallel to being opposite sides of rectangular block OABCDEFG. It is also given that is the centre of the square face CBFG and cm. Since is the centre of the square face CBFG, distance of point along from the origin is 3 units.
It is given that is parallel to and . It is evident from the diagram that is equal and parallel to being opposite sides of rectangular block OABCDEFG. It is evident from the diagram that distance of point along from the origin is 12 units.
It is given that is parallel to and . It is evident from the diagram that is equal and parallel to being opposite sides of rectangular block OABCDEFG. Since is the centre of the square face CBFG, distance of point along along from the origin is 3 units..
Hence, coordinates of .
Now we can represent the position vector of point as follows;
Now, let’s find coordinates of point . Consider the diagram.
It is given that is parallel to and it is evident from the diagram that distance of point along from the origin is ZERO.
It is given that is parallel to and . It is evident from the diagram that is equal and parallel to and being opposite sides of rectangular block OABCDEFG. Since it is given point is the midpoint of DG, distance of point along from the origin is 6 units.
It is given that is parallel to and . It is evident from the diagram that is equal and parallel to being opposite sides of rectangular block OABCDEFG. Therefore, distance of point along from the origin is 6 units.
Hence, coordinates of .
Now we can represent the position vector of point as follows;
Now have;
Therefore, we can write the vector as;
First let’s find vector .
A vector in the direction of is;
For the given case;
Therefore, we need the position vectors of points and .
A point has position vector from the origin . Then the position vector of is denoted by or .
Hence we need coordinates of the points and . We already have found the coordinates and position vector of point .
So, let’s find coordinates of point . Consider the diagram.
It is given that is parallel to and 6.
It is evident from the diagram that distance of point along from the origin is 6 units.
It is given that is parallel to and it is evident from the diagram that is parallel to being
opposite sides of rectangular block OABCDEFG. It is also given that AR = 4 cm. Hence, it is evident from the diagram that distance of point along from the origin is 4 units.
It is given that is parallel to and t is evident from the diagram that distance of point along along from the origin is ZERO.
Hence, coordinates of .
Now we can represent the position vector of point as follows;
Now have;
Therefore, we can write the vector as;
ii.
We recognize that angle RQP is between and . Therefore, we need the scalar/dot product of these two to find angle RQP.
From (i) we have;
The scalar or dot product of two vectors and in component form is given as;


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




Therefore, for the given case;
Therefore;
Hence;
Now we can equate the two equations of ;
Hence the angle RPQ is .
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