Final answer:
To find the area of a triangle using vectors, you need to find the cross product of two vectors representing two sides of the triangle. Using the given vertices, calculate the vectors AB and AC. Then, find the cross product of AB and AC, and calculate its magnitude to find the area of the triangle.
Step-by-step explanation:
To find the area of a triangle using vectors, you first need to find the two vectors representing two sides of the triangle. Then, you can use the cross product of those vectors to find the area. Let's call the given vertices A, B, and C. The vectors representing two sides of the triangle are AB and AC.
AB = B - A = (-2 - 1, 5 - 4, 1 - 6) = (-3, 1, -5)
AC = C - A = (1 - 1, -1 - 4, 1 - 6) = (0, -5, -5)
The cross product of these two vectors is given by: AB x AC = (-3, 1, -5) x (0, -5, -5)
Using the formula for the cross product, we find: AB x AC = (-15, -15, -15)
The magnitude of this cross product vector is the area of the triangle: |AB x AC| = sqrt((-15)^2 + (-15)^2 + (-15)^2) = 15 sqrt(3) square units.