Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2016  FebMar  (P19709/12)  Q#7
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Question
The diagram shows a pyramid OABC with a horizontal triangular base OAB and vertical height OC. Angles AOB, BOC and AOC are each right angles.
Unit vectors ,
i.Show that
ii.Express
iii.Use a scalar product to find angle BPC.
Solution
i.
It is evident from the diagram that from the side view we can see a right triangle AOC with angle OC given as right angle.
Pythagorean Theorem
For right triangle AOC;
It is evident from the diagram that;
We can also write it as;
Therefore;
Now we need to workout
A vector in the direction of
For the given case;
Therefore, we need the position vectors of points
A vector in the direction of
For the given case;
Therefore, we need the position vectors of points
To find
Now we need vector
·It is given that
·It is given that
·It is given that
Hence, coordinates of
Now we can represent the position vector of point
A point
To find
·It is given that
·It is given that
·It is given that
Hence, coordinates of
Now we can represent the position vector of point
A point
Now we can find
As we have seen above;
Therefore;
ii.
We are required to find
Let’s first find
It is evident from the diagram that;
We have seen in (i) that;
It is evident from the diagram that;
Therefore;
We can also write it as;
Therefore;
Hence;
We already have found in (i) that;
Hence;
Next we are required to find
It is evident from the diagram that;
We have found above that;
Let’s find
To find
·It is given that
·It is given that
·It is given that
Hence, coordinates of
Now we can represent the position vector of point
A point
Hence;
iii.
We are required to find the angle BPC.
It is evident from the diagram that angle BPC is between
Therefore, we use scalar/dot product of
From (i) and (ii) we have both
The scalar or dot product of two vectors


Since
Therefore for the given case;
The scalar or dot product of two vectors
where
For the given case;
Therefore;
Equating both scalar/dot products we get;
Hence the angle BPC is
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