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To solve this problem, we need to find a line \( L \) that is orthogonal to the two given lines and passes through the origin.

### Step 1: Identify the Direction Vectors
For the lines given in the problem:

1. **First line:**
   \[
   x = 4 + 4t, \quad y = 6 + 3t, \quad z = 7 - t
   \]
   The direction vector of this line, which we will call \( \vec{d}_1 \), can be found from the coefficients of \( t \):
   \[
   \vec{d}_1 = \langle 4, 3, -1 \rangle
   \]

2. **Second line:**
   \[
   x = 2 - 7s, \quad y = -1 + 5s, \quad z = 3 + 2s
   \]
   The direction vector of this line, \( \vec{d}_2 \), can be found from the coefficients of \( s \):
   \[
   \vec{d}_2 = \langle -7, 5, 2 \rangle
   \]

### Step 2: Find a Vector Orthogonal to Both Lines
A vector that is orthogonal to both \( \vec{d}_1 \) and \( \vec{d}_2 \) can be found by taking the cross product \( \vec{d}_1 \times \vec{d}_2 \).

\[
\vec{d}_1 \times \vec{d}_2 = 
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
4 & 3 & -1 \\
-7 & 5 & 2 \\
\end{vmatrix}
\]

Calculating this determinant:

\[
\vec{d}_1 \times \vec{d}_2 = \hat{i}(3 \cdot 2 - (-1) \cdot 5) - \hat{j}(4 \cdot 2 - (-1) \cdot (-7)) + \hat{k}(4 \cdot 5 - 3 \cdot (-7))
\]

\[
= \hat{i}(6 + 5) - \hat{j}(8 - 7) + \hat{k}(20 + 21)
\]

\[
= \hat{i}(11) - \hat{j}(1) + \hat{k}(41)
\]

So, the vector orthogonal to both \( \vec{d}_1 \) and \( \vec{d}_2 \) is:
\[
\vec{n} = \langle 11, -1, 41 \rangle
\]

### Step 3: Form the Equation of the Line \( L \)
Since we want the line \( L \) to pass through the origin, we can use the point \( (0, 0, 0) \) and the direction vector \( \vec{n} = \langle 11, -1, 41 \rangle \).

The parametric form of the line \( L \) is:
\[
x = 11t, \quad y = -t, \quad z = 41t
\]

### Final Answer
The equation of the line \( L \), which is orthogonal to both given lines and passes through the origin, is:
\[
x = 11t, \quad y = -t, \quad z = 41t
\]

Would you like a detailed explanation of any step?

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### Related Questions
1. How do we determine if two lines are orthogonal in 3D space?
2. How can the direction vector of a line help in determining line properties?
3. What is the significance of the cross product in finding orthogonal vectors?
4. Can the cross product be used to find the area of a parallelogram in 3D space?
5. How can we find the angle between two lines using their direction vectors?

#### Tip
The cross product of two vectors not only gives a vector orthogonal to both, but its magnitude represents the area of the parallelogram formed by the two vectors.

Encuentre una recta L ortogonal a las dos rectas dadas y que pase por el origen: x = 4 + 4t, y = 6 + 3t, z = 7 - t; x = 2 - 7s, y = -1 + 5s, z = 3 + 2s.

Learn how to find a line orthogonal to two given lines in 3D space that passes through the origin. This problem involves vector geometry concepts such as cross products and orthogonality. The solution includes detailed calculations for finding direction vectors and forming the parametric equation of the desired line.

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