To find the direction ratios of the normal to the plane passing through the origin and the points A(-2, 2, 3) and B(13, -3, 13), we can follow these steps:
### Step 1: Define the Points
We have two points:
- Point A: \( A(-2, 2, 3) \)
- Point B: \( B(13, -3, 13) \)
- The origin: \( O(0, 0, 0) \)
### Step 2: Find the Position Vectors
The position vectors from the origin to points A and B can be defined as:
- \( \vec{OA} = -2 \hat{i} + 2 \hat{j} + 3 \hat{k} \)
- \( \vec{OB} = 13 \hat{i} - 3 \hat{j} + 13 \hat{k} \)
### Step 3: Calculate the Cross Product
To find the direction ratios of the normal to the plane, we need to compute the cross product of the vectors \( \vec{OA} \) and \( \vec{OB} \).
The cross product \( \vec{OA} \times \vec{OB} \) can be calculated using the determinant of a matrix:
\[
\vec{OA} \times \vec{OB} =
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
-2 & 2 & 3 \\
13 & -3 & 13
\end{vmatrix}
\]
### Step 4: Calculate the Determinant
Expanding the determinant, we have:
\[
\vec{OA} \times \vec{OB} = \hat{i} \begin{vmatrix} 2 & 3 \\ -3 & 13 \end{vmatrix} - \hat{j} \begin{vmatrix} -2 & 3 \\ 13 & 13 \end{vmatrix} + \hat{k} \begin{vmatrix} -2 & 2 \\ 13 & -3 \end{vmatrix}
\]
Calculating each of these 2x2 determinants:
1. For \( \hat{i} \):
\[
2 \cdot 13 - 3 \cdot (-3) = 26 + 9 = 35
\]
2. For \( \hat{j} \):
\[
-(-2 \cdot 13 - 3 \cdot 13) = 26 - 39 = -13 \quad \text{(thus, it becomes +13)}
\]
3. For \( \hat{k} \):
\[
-2 \cdot (-3) - 2 \cdot 13 = 6 - 26 = -20
\]
Putting it all together, we get:
\[
\vec{OA} \times \vec{OB} = 35 \hat{i} + 13 \hat{j} - 20 \hat{k}
\]
### Step 5: Direction Ratios
The direction ratios of the normal vector to the plane are the coefficients of \( \hat{i}, \hat{j}, \hat{k} \):
- Direction ratios are \( 35, 13, -20 \).
### Step 6: Simplifying the Direction Ratios
We can simplify the direction ratios by taking out the common factor of 5:
\[
\text{Direction ratios} = 7, 13, -4
\]
### Final Answer
The direction ratios of the normal to the plane passing through the origin and the points A and B are \( 7, 13, -4 \).
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