2 Four points are given by their coordinatesin a rectangular

2. Four points are given by their coordinatesin a rectangular Cartesian system:

A(1,0,2); B(0,1,3); C(1,1,-1); and D(1,1,4)

(i) Evaluate the area of triangle ABD
(ii) Determine, if these four points are coplanar;
(iii) Evaluate the volume of the pyramid ABCD.

Solution

The sum is a problem of 3D Geometry, and not Algebra. However, still, I have tried to give some basic hints for the problem.

1) If a triangle and a parallelogram are on the same base and between the same parallels, then the area of triangle is equal to half the area of the parallelogram. Now, since, the triangle and the parallelogram are on the same base and between the same parallels, the area of triangle ABD is 1/2area of parallelogram. Solve it

2) Geometrically, the scalar triple product is the volume of the parallelepiped defined by the three vectors given. Now, by geometrical rules, ABCD will be coplanar if and only if u.(v X w)=0. Solve it

3) The volume of a pyramid is 1/3 × (the area of the base) × (the height). That is, volume of pyramid = (1/6) l u.(v x w) l. Thus, Volume of pyramid is 1/6 of volume of parallelepiped. Solve it

All the best.