Suppose a triangular shade sail is hung with its three vertices at (2,1, 9), (5,5,12) and (-3,4, 10).
Compute the area of the shade sail.

To compute the area of a triangular shade sail whose vertices are given in three-dimensional space, we can use the formula for the area of a triangle formed by three points \((A, B, C)\) in 3D space.

### Given vertices:
- \(A = (2, 1, 9)\)
- \(B = (5, 5, 12)\)
- \(C = (-3, 4, 10)\)

### Step-by-Step Solution:

1. **Vectors Representation**:
   Let's define two vectors from these three points:

   \[
   \overrightarrow{AB} = B - A = (5 - 2, 5 - 1, 12 - 9) = (3, 4, 3)
   \]
   \[
   \overrightarrow{AC} = C - A = (-3 - 2, 4 - 1, 10 - 9) = (-5, 3, 1)
   \]

2. **Cross Product**:
   The area of the triangle can be found using the magnitude of the cross product of these two vectors. The cross product \(\overrightarrow{AB} \times \overrightarrow{AC}\) is computed as:

   \[
   \overrightarrow{AB} \times \overrightarrow{AC} = 
   \begin{vmatrix}
   \mathbf{i} & \mathbf{j} & \mathbf{k} \\
   3 & 4 & 3 \\
   -5 & 3 & 1
   \end{vmatrix}
   \]

   The cross product determinant expands to:

   \[
   \overrightarrow{AB} \times \overrightarrow{AC} = \mathbf{i} (4 \cdot 1 - 3 \cdot 3) - \mathbf{j} (3 \cdot 1 - (-5) \cdot 3) + \mathbf{k} (3 \cdot 3 - (-5) \cdot 4)
   \]

   Simplify:

   \[
   \overrightarrow{AB} \times \overrightarrow{AC} = \mathbf{i} (4 - 9) - \mathbf{j} (3 + 15) + \mathbf{k} (9 + 20)
   \]
   \[
   = \mathbf{i} (-5) - \mathbf{j} (18) + \mathbf{k} (29)
   \]
   \[
   = (-5, -18, 29)
   \]

3. **Magnitude of the Cross Product**:
   The magnitude of the cross product is given by:

   \[
   \|\overrightarrow{AB} \times \overrightarrow{AC}\| = \sqrt{(-5)^2 + (-18)^2 + 29^2}
   \]

   Calculating this:

   \[
   = \sqrt{25 + 324 + 841}
   \]
   \[
   = \sqrt{1190}
   \]

4. **Area of the Triangle**:
   The area of the triangle is half the magnitude of the cross product:

   \[
   \text{Area} = \frac{1}{2} \times \sqrt{1190}
   \]
   
   Approximating the square root:

   \[
   \text{Area} \approx \frac{1}{2} \times 34.5 = 17.25
   \]

### Final Answer:
The area of the triangular shade sail is approximately **17.25 square units**.

Would you like more details, or do you have any questions?

### Related Questions:
1. How do you find the area of a triangle in 2D using vertex coordinates?
2. What is the geometric significance of the cross product in 3D space?
3. How can the vector cross product be used to determine the orientation of three points in 3D space?
4. What are other methods for calculating the area of a triangle in 3D space?
5. How do we compute the angle between two vectors in 3D space?

**Tip:** Always double-check the arithmetic when computing cross products and magnitudes, especially when dealing with negative numbers.

Suppose a triangular shade sail is hung with its three vertices at (2,1, 9), (5,5,12) and (-3,4,10). Compute the area of the shade sail.

Learn how to compute the area of a triangle in 3D space with vertices at (2,1,9), (5,5,12), and (-3,4,10). This problem demonstrates key vector algebra concepts, including cross products, and is suitable for students in Grades 10-12.

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