use the cross product to find the area of the triangle with vertices (a,0,0) (0,b,0) and (0,0,c)

The area of a triangle with vertices \((a, 0, 0)\), \((0, b, 0)\), and \((0, 0, c)\) can be found using the cross product of two vectors representing two sides of the triangle.

### Step-by-Step Process:

1. **Define the vectors**:
   - Let \(\mathbf{v}_1\) be the vector from \((a, 0, 0)\) to \((0, b, 0)\):
     \[
     \mathbf{v}_1 = (0 - a, b - 0, 0 - 0) = (-a, b, 0)
     \]
   - Let \(\mathbf{v}_2\) be the vector from \((a, 0, 0)\) to \((0, 0, c)\):
     \[
     \mathbf{v}_2 = (0 - a, 0 - 0, c - 0) = (-a, 0, c)
     \]

2. **Compute the cross product** \(\mathbf{v}_1 \times \mathbf{v}_2\):
   \[
   \mathbf{v}_1 \times \mathbf{v}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -a & b & 0 \\ -a & 0 & c \end{vmatrix}
   \]
   Expanding the determinant:
   \[
   \mathbf{v}_1 \times \mathbf{v}_2 = \mathbf{i} \left(bc - 0\right) - \mathbf{j} \left(ac - 0\right) + \mathbf{k} \left(0 - (-a)(b)\right)
   \]
   Simplifying:
   \[
   \mathbf{v}_1 \times \mathbf{v}_2 = (bc, -ac, ab)
   \]

3. **Find the magnitude of the cross product**:
   The magnitude of \(\mathbf{v}_1 \times \mathbf{v}_2\) is:
   \[
   |\mathbf{v}_1 \times \mathbf{v}_2| = \sqrt{(bc)^2 + (-ac)^2 + (ab)^2}
   \]
   Simplifying:
   \[
   |\mathbf{v}_1 \times \mathbf{v}_2| = \sqrt{b^2c^2 + a^2c^2 + a^2b^2}
   \]
   Factor out the common terms:
   \[
   |\mathbf{v}_1 \times \mathbf{v}_2| = \sqrt{a^2b^2 + a^2c^2 + b^2c^2}
   \]

4. **Area of the triangle**:
   The area of the triangle is half the magnitude of the cross product:
   \[
   \text{Area} = \frac{1}{2} |\mathbf{v}_1 \times \mathbf{v}_2| = \frac{1}{2} \sqrt{a^2b^2 + a^2c^2 + b^2c^2}
   \]

Thus, the area of the triangle with vertices \((a, 0, 0)\), \((0, b, 0)\), and \((0, 0, c)\) is:

\[
\boxed{\frac{1}{2} \sqrt{a^2b^2 + a^2c^2 + b^2c^2}}
\]

Would you like further details or have any questions?

Here are 5 related questions for practice:
1. What is the cross product of \(\mathbf{v}_1 = (1, 2, 0)\) and \(\mathbf{v}_2 = (0, -1, 3)\)?
2. How can the cross product be used to find the area of a parallelogram?
3. What is the geometrical interpretation of the magnitude of the cross product?
4. How would you find the area of a triangle in 3D if one of the vertices is at the origin?
5. What happens to the area if two of the vertices coincide?

**Tip:** The cross product is always perpendicular to the plane containing the two vectors involved, making it a useful tool for 3D geometry.

Learn how to calculate the area of a triangle with vertices (a, 0, 0), (0, b, 0), and (0, 0, c) using the cross product method. This solution involves vector algebra, the cross product formula, and geometric interpretation in 3D space. Ideal for high school students in Grades 10-12.

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