How can i find the area of the triangle vertices (2,-3,1), (1,-1,2)&(-1,2,3)
To find the area of a triangle given its vertices, you can use the formula for the area of a triangle in three-dimensional space.
The formula is as follows:
Area = 1/2 * |(x2 - x1) * (y3 - y1) * (z3 - z1) + (y2 - y1) * (z3 - z1) * (x3 - x1) + (z2 - z1) * (x3 - x1) * (y3 - y1) - (z2 - z1) * (y3 - y1) * (x3 - x1) - (y2 - y1) * (x3 - x1) * (z3 - z1) - (x2 - x1) * (z3 - z1) * (y3 - y1)|
Using the coordinates of the three vertices:
A(2, -3, 1), B(1, -1, 2), and C(-1, 2, 3),
you can substitute these values into the formula and calculate the area.
Here's the step-by-step calculation:
1. Calculate the differences between the x, y, and z coordinates of the three vertices:
(x2 - x1) = 1 - 2 = -1
(y2 - y1) = -1 - (-3) = 2
(z2 - z1) = 2 - 1 = 1
(x3 - x1) = -1 - 2 = -3
(y3 - y1) = 2 - (-3) = 5
(z3 - z1) = 3 - 1 = 2
2. Plug the differences into the formula and calculate the area:
Area = 1/2 * |(-1) * 5 * 2 + 2 * 2 * (-3) + 1 * (-3) * 2 - 1 * 2 * (-3) - 2 * (-3) * 5 - (-1) * 2 * 2|
= 1/2 * |(-10) + (-12) + (-6) + 6 + 30 - 8|
= 1/2 * |0|
= 0
Therefore, the area of the triangle with vertices (2, -3, 1), (1, -1, 2), and (-1, 2, 3) is 0.