Let P=(0,2,1)
, Q=(2,0,1)
 and R=(3,−1,2)
 . The area of the triangle determined by the points P
 ,  Q
, and R
 is

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:

\[
\vec{PQ} \times \vec{PR} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
2 & -2 & 0 \\
3 & -3 & 1
\end{vmatrix}
\]

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.

Learn how to calculate the area of a triangle using vectors and the cross product method with points (0, 2, 1), (2, 0, 1), and (3, -1, 2). This problem involves applying vector mathematics and geometric concepts. Suitable for high school students studying vectors and geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation