Using vectors, find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5), and C(1, 5, 5).

To find the area of the triangle formed by the vertices A(1, 1, 2), B(2, 3, 5), and C(1, 5, 5) using vectors, we can follow these steps:

Step 1: Define the Vectors

First, we need to create two vectors from the points:

• Vector AB = B - A = (2 - 1, 3 - 1, 5 - 2) = (1, 2, 3)
• Vector AC = C - A = (1 - 1, 5 - 1, 5 - 2) = (0, 4, 3)

Step 2: Calculate the Cross Product

The area of the triangle can be found using the cross product of vectors AB and AC. The formula for the cross product of two vectors (a1, a2, a3) and (b1, b2, b3) is:

Cross Product = (a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1)

Applying this to our vectors:

• AB = (1, 2, 3)
• AC = (0, 4, 3)

Cross Product = (2*3 - 3*4, 3*0 - 1*3, 1*4 - 2*0) = (6 - 12, 0 - 3, 4 - 0) = (-6, -3, 4)

Step 3: Find the Magnitude of the Cross Product

The magnitude of the cross product vector gives us twice the area of the triangle:

Magnitude = √((-6)² + (-3)² + 4²) = √(36 + 9 + 16) = √61

Step 4: Calculate the Area

The area of the triangle is half the magnitude of the cross product:

Area = 1/2 * √61

Final Result

The area of the triangle with vertices A(1, 1, 2), B(2, 3, 5), and C(1, 5, 5) is:

Area = 1/2 * √61