Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2016  OctNov  (P19709/13)  Q#7
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Question
The diagram shows a triangular pyramid ABCD. It is given that



i. Verify, showing all necessary working, that each of the angles DAB, DAC and CAB is 90^{o}.
ii. Find the exact value of the area of the triangle ABC, and hence find the exact value of the volume of the pyramid.
[The volume V of a pyramid of base area A and vertical height h is given by ]
Solution
i.
We recognize that angle DAB is between and .
We are given that DAB is a 90^{o}.
If and & , then and are perpendicular.
Therefore;
Now, we need the scalar/dot product of and .
The scalar or dot product of two vectors and in component form is given as;


Since ;


Hence, and are perpendicular and therefore, angle DAB is 90^{o}.
We recognize that angle DAC is between and .
We are given that DAC is a 90^{o}.
If and & , then and are perpendicular.
Therefore;
Now, we need the scalar/dot product of and .
The scalar or dot product of two vectors and in component form is given as;


Since ;


Hence, and are perpendicular and therefore, angle DAC is 90^{o}.
We recognize that angle CAB is between and .
We are given that CAB is a 90^{o}.
If and & , then and are perpendicular.
Therefore;
Now, we need the scalar/dot product of and .
The scalar or dot product of two vectors and in component form is given as;


Since ;


Hence, and are perpendicular and therefore, angle CAB is 90^{o}.
ii.
We are required to find area of triangle ABC. As shown in (i), angle CAB is 90^{o}.
Therefore ABC is a right angle triangle.
Expression for the area of the triangle is;
For triangle ABC;
We are given that;


Therefore, we need to find magnitudes (lengths) of these vectors to find area of triangle ABC.
Expression for the length (magnitude) of a vector is;






Therefore;
Now we need to find volume of pyramid.
The volume V of a pyramid of base area A and vertical height h is given by;
We have found area of triangle (ABC) i.e. base of pyramid. We need height of the pyramid to find the volume of pyramid.
It is evident from the diagram that height of the pyramid is AD.
Therefore, we need to find magnitude (length) of this vector to find height of pyramid.
Expression for the length (magnitude) of a vector is;
We are given;
Hence, volume of pyramid is;
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