https://file.aiquickdraw.com/mathbot/file/17285238871723tlkz4y1.jpg

To find the area of the triangle formed by three vertices in 3D space, we can use the **cross product** of two vectors from these vertices. The formula for the area of a triangle with vertices \( A(x_1, y_1, z_1), B(x_2, y_2, z_2), C(x_3, y_3, z_3) \) is:

\[
\text{Area} = \frac{1}{2} \left| \vec{AB} \times \vec{AC} \right|
\]

Where \( \vec{AB} \) and \( \vec{AC} \) are the vectors from point \( A \) to points \( B \) and \( C \), respectively.

### Step 1: Calculate the vectors \( \vec{AB} \) and \( \vec{AC} \)
- \( A = (0, 4, 4) \)
- \( B = (2, -6, -5) \)
- \( C = (-3, -5, 6) \)

\[
\vec{AB} = B - A = (2 - 0, -6 - 4, -5 - 4) = (2, -10, -9)
\]
\[
\vec{AC} = C - A = (-3 - 0, -5 - 4, 6 - 4) = (-3, -9, 2)
\]

### Step 2: Find the cross product \( \vec{AB} \times \vec{AC} \)
The cross product of two vectors \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) is given by:

\[
\vec{AB} \times \vec{AC} = \left( y_1 z_2 - z_1 y_2, z_1 x_2 - x_1 z_2, x_1 y_2 - y_1 x_2 \right)
\]

Substituting \( \vec{AB} = (2, -10, -9) \) and \( \vec{AC} = (-3, -9, 2) \):

\[
\vec{AB} \times \vec{AC} = \left( (-10)(2) - (-9)(-9), (-9)(-3) - 2(2), 2(-9) - (-10)(-3) \right)
\]
\[
\vec{AB} \times \vec{AC} = (-20 - 81, 27 - 4, -18 - 30)
\]
\[
\vec{AB} \times \vec{AC} = (-101, 23, -48)
\]

### Step 3: Calculate the magnitude of the cross product
The magnitude of \( \vec{AB} \times \vec{AC} \) is:

\[
|\vec{AB} \times \vec{AC}| = \sqrt{(-101)^2 + 23^2 + (-48)^2}
\]
\[
|\vec{AB} \times \vec{AC}| = \sqrt{10201 + 529 + 2304} = \sqrt{13034}
\]
\[
|\vec{AB} \times \vec{AC}| \approx 114.14
\]

### Step 4: Calculate the area of the triangle
The area of the triangle is:

\[
\text{Area} = \frac{1}{2} \times 114.14 \approx 57.07
\]

Thus, the area of the triangle is approximately **57.07 square units**.

Let me know if you'd like further details or explanations.

### Related questions:
1. How do you calculate the area of a triangle in 2D versus 3D?
2. What is the significance of the cross product in determining the area?
3. Can the same formula be used for quadrilaterals or polygons in 3D?
4. What are the properties of the vectors formed by vertices in space?
5. How can we determine if three points form a right triangle in 3D?

### Tip:
When working with cross products, always remember that the order of vectors matters, as the cross product is not commutative (i.e., \( \vec{A} \times \vec{B} \neq \vec{B} \times \vec{A} \)).

Find the area of the triangle formed by the three vertices A = (0, 4, 4), B = (2, -6, -5), and C = (-3, -5, 6).

Learn how to find the area of a triangle in 3D space formed by vertices A(0, 4, 4), B(2, -6, -5), and C(-3, -5, 6) using the cross product. The problem involves calculating vectors, performing a cross product, and finding the magnitude of the resulting vector.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation