Question

A building is to be constructed in the form of a triangular pyramid ABCD as shown in the figure. Let the angular points be A(0, 1, 2), B(3, 0, 1), C(4, 3, 6) and D(2, 3, 2) and let G be the point of intersection of the medians of ∆BCD.

Using the above information, answer the following:

1. What will be the length of vector `vec(AG)`?
2. Find the area of ∆ABC.

Answer 1

a. G is the centroid of ∆BCD.

The coordinates are

`((3 + 4 + 2)/3, (0 + 3 + 3)/3, (1 + 6 + 2)/3)`

= (3, 2, 3)

`vec(AG) = (3 - 0)hati + (2 - 1)hatj + (3 - 2)hatk`

`vec(AG) = 3hati + hatj + hatk`

`|vec(AG)| = sqrt(3^2 + 1^2 + 1^2)`

`|vec(AG)| = sqrt(11)` units

b. `vec(AB) = (3 - 0)hati + (0 - 1)hatj + (1 - 2)hatk`

`vec(AB) = 3hati - hatj - hatk`

`vec(AC) = (4 - 0)hati + (3 - 1)hatj + (6 - 2)hatk`

`vec(AC) = 4hati + 2hatj + 4hatk`

Area of ∆ABC = `1/2|vec(AB) xx vec(AC)|`

`vec(AB) xx vec(AC) = |(hati, hatj, hatk),(3, -1, -1),(4, 2, 4)|`

`vec(AB) xx vec(AC) = 2hati - 16hatj + 10hatk` and `|vec(AB) xx vec(AC)| = 6sqrt(10)`

Hence area of ∆ABC = `1/2 |vec(AB) xx vec(AC)| = 3sqrt(10)` sq units.