To Find the Area of the Triangle with vertices A(1,2,3), B(2,3,4) and C(3,4,5):
Step 1: Use the Formula
Area of triangle = (1/2) |AB × AC|
First, find vectors AB and AC.
AB = B − A = (2−1, 3−2, 4−3) = (1,1,1)
AC = C − A = (3−1, 4−2, 5−3) = (2,2,2)
Step 2: Compute Cross Product AB × AC
AB × AC = | i j k |
| 1 1 1 |
| 2 2 2 |
Expanding determinant:
= i(1×2 − 1×2) − j(1×2 − 1×2) + k(1×2 − 1×2)
= i(2−2) − j(2−2) + k(2−2)
= 0i − 0j + 0k = (0,0,0)
Step 3: Find Magnitude
|AB × AC| = 0
Therefore, Area = (1/2) × 0 = 0
Conclusion:
The area of the triangle is 0.
This means the three points are collinear (they lie on a straight line).