Complete the parametric equations of the line through the point (6,0,-4) and parallel to the vector
⟨
4
,
1
,
−
6
⟩

To write the parametric equations of the line passing through the point \((6, 0, -4)\) and parallel to the vector \(\langle 4, 1, -6 \rangle\), we follow the general formula for parametric equations of a line:

\[
\text{Point on the line: } (x_0, y_0, z_0)
\]
\[
\text{Direction vector: } \langle a, b, c \rangle
\]
The parametric equations are given by:

\[
x = x_0 + at
\]
\[
y = y_0 + bt
\]
\[
z = z_0 + ct
\]

### Given:
- Point \((x_0, y_0, z_0) = (6, 0, -4)\)
- Direction vector \(\langle a, b, c \rangle = \langle 4, 1, -6 \rangle\)

### Parametric Equations:
\[
x = 6 + 4t
\]
\[
y = 0 + 1t \quad \text{or} \quad y = t
\]
\[
z = -4 - 6t
\]

Thus, the parametric equations for the line are:
\[
x = 6 + 4t, \quad y = t, \quad z = -4 - 6t
\]

Would you like more details or have any questions? 

Here are 5 related questions for further exploration:
1. How do we find parametric equations if the line is perpendicular to a vector?
2. What are the parametric equations of a line through two given points?
3. How can we convert parametric equations into symmetric equations of a line?
4. How do you determine if two parametric lines are parallel or coincident?
5. How do we find the point of intersection of two parametric lines?

**Tip:** Parametric equations are useful in describing lines in 3D space, especially for motion and vector analysis.

Complete the parametric equations of the line through the point (6,0,-4) and parallel to the vector ⟨4,1,−6⟩.

Discover how to write the parametric equations for a line passing through the point (6,0,-4) and parallel to the vector ⟨4,1,−6⟩. This solution covers key concepts in analytic geometry, such as parametric equations and vector directions. Suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation