Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | 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 90o.

   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 90o.

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 90o.

We recognize that angle DAC is between  and .

We are given that DAC is a 90o.

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 90o.

We recognize that angle CAB is between  and .

We are given that CAB is a 90o.

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 90o.

 

   ii.
 

We are required to find area of triangle ABC. As shown in (i), angle CAB is 90o.

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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