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 centroid or barycenter of a set of points. The coordinates of the centroid are the average of the coordinates of the points.
Given that the coordinates of the barycenter of {A, B, C} are (4/3, 1/3, 4/3), let's assume the coordinates of point B are (x, y, z).
To find the values of x, y, and z, we need to use the formula for finding the centroid:
x = (xA + xB + xC) / 3
y = (yA + yB + yC) / 3
z = (zA + zB + zC) / 3
We know the coordinates of points A and C, so we can substitute those values in the equation:
1) x = (1 + xB + 2) / 3
2) y = (1 + yB + 0) / 3
3) z = (1 + zB + 1) / 3
Let's simplify each equation:
1) 3x = 3 + xB + 2
2) 3y = 1 + yB + 0
3) 3z = 3 + zB + 1
Rewrite the equations:
1) x = 5 - xB
2) y = 1 + yB
3) z = 4 - zB
Now, we can substitute the coordinates of the centroid (4/3, 1/3, 4/3) into these equations:
1) 4/3 = 5 - xB
2) 1/3 = 1 + yB
3) 4/3 = 4 - zB
Solve each equation for xB, yB, and zB:
1) xB = 5 - 4/3
2) yB = 1/3 - 1
3) zB = 4 - 4/3
Simplify:
1) xB = 11/3
2) yB = -2/3
3) zB = 8/3
Therefore, the coordinates of point B in the given spot are (11/3, -2/3, 8/3).