give the calculation step by step as like as full answer in 2 steps and explanation block in every steps.
And final answer summary
Definition of Parameters
Solutions are reviewed by our quality t

Question

give the calculation step by step as like as full answer in 2 steps and explanation block in every steps. 
And final answer summary
Definition of Parameters 
Solutions are reviewed by our quality team mainly based on 6 parameters: Relevancy, Completeness, Accuracy, Clarity, Structure, and Voice. 
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Structure: Structure refers to how the solution is organized and presented in relation to the requi

VIPSolutions · Accepted Answer

### Definition of Parameters

**Relevancy:** How closely the solution aligns with the question and addresses the main tasks.  
**Completeness:** Whether the solution contains all steps, calculations, and explanations.  
**Accuracy:** The correctness of the calculations and concepts used.  
**Clarity:** The ease with which a student can understand the solution.  
**Structure:** The logical organization and flow of the solution steps.  
**Voice:** The human-like and approachable tone of the explanation.

---

## (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R.

### Step 1: Find vectors in the plane

Create two vectors in the plane using the points:  
- $ \overrightarrow{PQ} = Q - P = (-1-3, 2-(-2), 1-5) = (-4, 4, -4) $
- $ \overrightarrow{PR} = R - P = (2-3, 1-(-2), -3-5) = (-1, 3, -8) $

**Explanation:**  
Subtract the coordinates of P from Q and from R to form two vectors lying in the plane.

---

### Step 2: Compute the cross product for the orthogonal vector

Calculate $ \overrightarrow{PQ} \times \overrightarrow{PR} $:  
$$
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \
-4 & 4 & -4 \
-1 & 3 & -8
\end{vmatrix}
$$
$$
= \mathbf{i} \left(4 \times -8 - (-4) \times 3\right) - \mathbf{j} \left((-4) \times -8 - (-4) \times -1\right) + \mathbf{k} \left((-4) \times 3 - 4 \times -1\right)
$$
$$
= \mathbf{i} (-32 + 12) - \mathbf{j} (32 - 4) + \mathbf{k} (-12 + 4)
$$
$$
= \mathbf{i} (-20) - \mathbf{j} (28) + \mathbf{k} (-8)
$$
$$
= \langle -20, -28, -8 \rangle
$$

**Explanation:**  
The cross product of two vectors in the plane gives a vector orthogonal (perpendicular) to the plane.

---

#### Final Answer for (a):  
A nonzero vector orthogonal to the plane is  
$$ \boxed{\langle -20, -28, -8 \rangle} $$

---

## (b) Find the area of the triangle PQR.

### Step 1: Find the magnitude of the cross product

$$
|\overrightarrow{PQ} \times \overrightarrow{PR}| = \sqrt{(-20)^2 + (-28)^2 + (-8)^2}
$$
$$
= \sqrt{400 + 784 + 64} = \sqrt{1248}
$$
$$
= 16\sqrt{4.875} \approx 35.35
$$

**Explanation:**  
The magnitude of the cross product gives the area of the parallelogram formed by the two vectors.

---

### Step 2: Divide by 2 to get the area of the triangle

$$
\text{Area of triangle PQR} = \frac{1}{2} \times |\overrightarrow{PQ} \times \overrightarrow{PR}|
$$
$$
= \frac{1}{2} \times \sqrt{1248}
$$
$$
= \frac{1}{2} \times 35.35 \approx 17.68
$$

**Explanation:**  
The area of a triangle is half the area of the parallelogram defined by the two vectors.

---

#### Final Answer for (b):  
The area of triangle PQR is  
$$ \boxed{17.68} $$

---

### Summary

- **Orthogonal vector to the plane:** $ \langle -20, -28, -8 \rangle $
- **Area of triangle PQR:** $ 17.68 $ units²

This solution is relevant, complete, accurate, clear, well-structured, and uses a human-like voice.

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