4 points A,B,C and D are situated in a 3-dimensional space. In a certain spot, the coordinates of A, C and D are known:
A(1,1,1)
C(2,0,1)
D(0,3,0)
The coordinates of the barycentre of {A,B,C} are: (4/3, 1/3, 4/3).
a) Determine the coordinates of point B in the used spot.
To determine the coordinates of point B, we can use the concept of the barycentre, which is the weighted average of the coordinates of the points in consideration.
Given that the barycentre coordinates of {A, B, C} are (4/3, 1/3, 4/3), we can set up the following equation based on the definition of the barycentre:
(4/3)*(coordinates of A) + (1/3)*(coordinates of B) + (4/3)*(coordinates of C) = (coordinates of barycentre)
Substituting the given coordinates:
(4/3)*(1, 1, 1) + (1/3)*(coordinates of B) + (4/3)*(2, 0, 1) = (4/3, 1/3, 4/3)
Simplifying the equation:
(4/3 + 8/3, 4/3 + 0, 4/3 + 4/3) + (1/3)*(coordinates of B) = (4/3, 1/3, 4/3)
Combining like terms:
(12/3, 4/3, 12/3) + (1/3)*(coordinates of B) = (4/3, 1/3, 4/3)
Simplifying further:
(4, 4/3, 4) + (1/3)*(coordinates of B) = (4/3, 1/3, 4/3)
Now, we can equate the corresponding coordinates and solve for point B:
4 + (1/3)*Bx = 4/3
4/3 + (1/3)*By = 1/3
4 + (1/3)*Bz = 4/3
Solving these equations, we find that:
Bx = 1/9
By = -1/9
Bz = 5/3
Therefore, the coordinates of point B in the given spot are (1/9, -1/9, 5/3).