Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2018  MayJun  (P19709/13)  Q#9
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Question
The diagram shows a pyramid OABCD with a horizontal rectangular base OABC. The sides OA and AB have lengths of 8 units and 6 units respectively. The point E on OB is such that OE = 2 units. The point D of the pyramid is 7 units vertically above E. Unit vectors , and are parallel to , and respectively.
i. Show that .
ii. Use a scalar product to find angle BDO.
Solution
i.
We are given length / magnitude of as;
We are required to find as given.
A unit vector in the direction of is;
Therefore, we can write that a vector with given length / magnitude and the direction of a given vector / unit vector as;
Hence, for ;
We are already given but we need to find .
To find consider the diagram below.
It is evident from the diagram that is parallel to .
Therefore, unit vector of and is same.
Hence, first we find unit vector of .
A unit vector in the direction of is;
Therefore, to find the unit vector of we need length / magnitude and vector .
First we find and then its length / magnitude.
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 8 units. (OABC being a rectangular base)
· It is given that is parallel to ( and hence, , OABC being a rectangular base) and we can see that distance of point along from the origin is 6. Hence, distance of of point along from the origin is also 6.
· It is given that is parallel to and we can see that distance of point along from the origin is ZERO.
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 need to find the unit vector of .
Hence, unit vector of can be found as;
We know that same is the unit vector of .
Finally, we can find the vector .
ii.
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 ;
Therefore, we need to find and .
First let us find the vector . Consider the diagram below.
It is evident from the diagram that;
From (i), we have;
Next we find .
We are given that from point to , the distance is covered only vertically, along and is 7 units, and no distance is covered along and .
Hence;
Therefore;
Next we need to find .
Consider the diagram below.
From (i), we have;
Therefore;
We have also found above that;
Hence;
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 BDO is .
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