Give a vector parametric equation for the line through the point (−4,5,3) that is parallel to the line ⟨1−3t,−4−t,0⟩:

To find the parametric equation of a line that passes through the point \((-4, 5, 3)\) and is parallel to the line given by \(\langle 1 - 3t, -4 - t, 0 \rangle\), follow these steps:

1. **Identify the direction vector**: The parametric equation of the given line \(\langle 1 - 3t, -4 - t, 0 \rangle\) has a direction vector \(\mathbf{d} = \langle -3, -1, 0 \rangle\), which represents the direction of the line.

2. **Set up the parametric equation**: A parametric equation for a line passing through a point \((x_0, y_0, z_0)\) with direction vector \(\mathbf{d} = \langle a, b, c \rangle\) is:
   \[
   \mathbf{r}(t) = \langle x_0, y_0, z_0 \rangle + t \langle a, b, c \rangle
   \]

   In this case, the point is \((-4, 5, 3)\), and the direction vector is \(\langle -3, -1, 0 \rangle\).

3. **Write the parametric equation**:
   \[
   \mathbf{r}(t) = \langle -4, 5, 3 \rangle + t \langle -3, -1, 0 \rangle
   \]

   This simplifies to:
   \[
   \mathbf{r}(t) = \langle -4 - 3t, 5 - t, 3 \rangle
   \]

Thus, the parametric equation of the line is:
\[
\mathbf{r}(t) = \langle -4 - 3t, 5 - t, 3 \rangle
\]

### Do you have any questions, or would you like more details?

Here are some related questions:
1. How do you find the parametric equation of a line given two points?
2. What is the vector equation of a line, and how does it differ from the parametric form?
3. How do you calculate the angle between two lines using their direction vectors?
4. What does it mean geometrically for two lines to be parallel in 3D space?
5. How do you find the point of intersection between two lines in space?

**Tip**: Always check the direction vector to confirm whether lines are parallel, perpendicular, or skew.

Give a vector parametric equation for the line through the point (−4,5,3) that is parallel to the line ⟨1−3t,−4−t,0⟩.

Learn how to derive the parametric equation for a line passing through the point (-4, 5, 3) and parallel to the line ⟨1 - 3t, -4 - t, 0⟩. This tutorial covers the concepts of vector geometry and parametric equations, suitable for high school students in Grades 10-12.

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