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.