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For problem 10, we are given:

Point \( M_0(-1, -4, 5) \) and a direction vector \( \vec{I} = \begin{bmatrix} -1 \\ 0 \\ 7 \end{bmatrix} \). We need to write the parametric and canonical equations of the line passing through point \( M_0 \) and parallel to \( \vec{I} \).

### Solution:

1. **Parametric Equations:**
   The parametric form of a line that passes through point \( M_0(x_0, y_0, z_0) \) and is parallel to direction vector \( \vec{I} = \begin{bmatrix} a \\ b \\ c \end{bmatrix} \) is:
   \[
   x = x_0 + at, \quad y = y_0 + bt, \quad z = z_0 + ct
   \]
   Substituting the values:
   - \( x_0 = -1 \), \( y_0 = -4 \), \( z_0 = 5 \)
   - \( a = -1 \), \( b = 0 \), \( c = 7 \)

   So, the parametric equations become:
   \[
   x = -1 - t
   \]
   \[
   y = -4
   \]
   \[
   z = 5 + 7t
   \]

2. **Canonical Form:**
   The canonical form is derived by eliminating the parameter \( t \) from the parametric equations. For the given line:
   \[
   \frac{x + 1}{-1} = \frac{y + 4}{0} = \frac{z - 5}{7}
   \]
   
   Since \( \frac{y + 4}{0} \) is undefined, this implies \( y = -4 \) for all points on the line.

   Therefore, the canonical equation is:
   \[
   \frac{x + 1}{-1} = \frac{z - 5}{7}, \quad y = -4
   \]

### Final Answer:
- **Parametric Equations:** \( x = -1 - t \), \( y = -4 \), \( z = 5 + 7t \)
- **Canonical Equation:** \( \frac{x + 1}{-1} = \frac{z - 5}{7}, \quad y = -4 \)

Would you like more details or have any questions?

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Here are some related questions that may help deepen your understanding:

1. How would you write the equations if the direction vector had a zero component in the \( z \)-direction?
2. What does it mean for a line to be parallel to a vector in three-dimensional space?
3. How can we determine if two lines in space are parallel or intersect?
4. How does the direction vector influence the slope of the line in space?
5. Can the parametric form represent lines in higher dimensions?

**Tip:** For lines in 3D space, a zero component in the direction vector implies that the line does not vary along that axis, which simplifies the equations.

Solve problem 10: Write the parametric and canonical equations of the line passing through point M_0(-1, -4, 5) and parallel to vector I = [-1, 0, 7].

Learn how to find the parametric and canonical equations of a line passing through point M_0(-1, -4, 5) and parallel to vector I = [-1, 0, 7]. This solution covers vector algebra concepts, including parametric and canonical equations in 3D. Ideal for high school students in Grades 10-12.

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