To find the area of the triangle determined by the points $P = (0, 2, 1)$, $Q = (2, 0, 1)$, and $R = (3, -1, 2)$, we can use the cross product of the vectors $\vec{PQ}$ and $\vec{PR}$ formed by these points.

Step 1: Find the vectors $\vec{PQ}$ and $\vec{PR}$

$\vec{PQ} = \overrightarrow{Q} - \overrightarrow{P} = (2 - 0, 0 - 2, 1 - 1) = (2, -2, 0)$

$\vec{PR} = \overrightarrow{R} - \overrightarrow{P} = (3 - 0, -1 - 2, 2 - 1) = (3, -3, 1)$

Step 2: Compute the cross product $\vec{PQ} \times \vec{PR}$

The cross product of $\vec{PQ} = (2, -2, 0)$ and $\vec{PR} = (3, -3, 1)$ is calculated as follows:

\hat{i} & \hat{j} & \hat{k} \\ 2 & -2 & 0 \\ 3 & -3 & 1 \end{vmatrix}$$ Expanding this determinant: $$\vec{PQ} \times \vec{PR} = \hat{i} \left((-2)(1) - (0)(-3)\right) - \hat{j} \left((2)(1) - (0)(3)\right) + \hat{k} \left((2)(-3) - (-2)(3)\right)$$ $$\vec{PQ} \times \vec{PR} = \hat{i}(-2) - \hat{j}(2) + \hat{k}(0) = (-2\hat{i} - 2\hat{j} + 0\hat{k})$$ So, $$\vec{PQ} \times \vec{PR} = (-2, -2, 0)$$ ### Step 3: Find the magnitude of the cross product The magnitude of the cross product $$\vec{PQ} \times \vec{PR}$$ gives twice the area of the triangle: $$|\vec{PQ} \times \vec{PR}| = \sqrt{(-2)^2 + (-2)^2 + 0^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$$ ### Step 4: Calculate the area of the triangle The area $$A$$ of the triangle is half the magnitude of the cross product: $$A = \frac{1}{2} |\vec{PQ} \times \vec{PR}| = \frac{1}{2} \times 2\sqrt{2} = \sqrt{2}$$ Thus, the area of the triangle is $$\sqrt{2}$$ square units. Would you like more details or have any questions? --- Here are 5 related questions: 1. How do you find the volume of a parallelepiped formed by three vectors? 2. What is the significance of the scalar triple product in vector calculus? 3. Can you derive the formula for the area of a parallelogram using vector cross product? 4. How does the orientation of vectors affect the sign of their cross product? 5. What are some real-life applications of finding areas using vectors? **Tip:** Remember that the magnitude of the cross product of two vectors gives the area of the parallelogram they span. For a triangle, it's half of that area.