55. Write parametric equations of each given line:
a) the line in R
3 passing through (11, −6, 7) parallel to vector 3i + j − 12k
b) the line in R
3 passing through the points (3, 1, 3) and (2, −4, −2)
c) the x-axis (in R
2
)
56. Write parametric equations of each given line:
a) the line in R
4 passing through the points (−1, 0, 3, −12) and (6, −15, 11, −8)
b) the line in R
2 whose slope-intercept equation is y =
−1
3
x −
5
3
c) the x-axis (in R
3
)

Let's tackle the problem by writing parametric equations for each of the lines as described.

### **55. Parametric Equations in \( \mathbb{R}^3 \):**

#### **(a) Line passing through \((11, -6, 7)\) and parallel to the vector \(3\mathbf{i} + \mathbf{j} - 12\mathbf{k}\):**

The parametric equation of a line in \( \mathbb{R}^3 \) is given by:

\[
\mathbf{r}(t) = \mathbf{r}_0 + t \mathbf{v}
\]

Where:
- \( \mathbf{r}_0 = (x_0, y_0, z_0) \) is a point on the line, and
- \( \mathbf{v} = (v_1, v_2, v_3) \) is the direction vector of the line.
  
Here, \( \mathbf{r}_0 = (11, -6, 7) \) and \( \mathbf{v} = (3, 1, -12) \). Thus, the parametric equations of the line are:

\[
x = 11 + 3t
\]
\[
y = -6 + t
\]
\[
z = 7 - 12t
\]

#### **(b) Line passing through the points \((3, 1, 3)\) and \((2, -4, -2)\):**

To find the parametric equation of a line passing through two points, we first need to determine the direction vector:

\[
\mathbf{v} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)
\]

Substituting the given points \((3, 1, 3)\) and \((2, -4, -2)\):

\[
\mathbf{v} = (2 - 3, -4 - 1, -2 - 3) = (-1, -5, -5)
\]

Now, the parametric equations of the line through the point \((3, 1, 3)\) are:

\[
x = 3 - t
\]
\[
y = 1 - 5t
\]
\[
z = 3 - 5t
\]

#### **(c) Line along the x-axis in \( \mathbb{R}^2 \):**

For a line along the x-axis in \( \mathbb{R}^2 \), the \( y \)-coordinate is always 0, and \( x \) varies freely. The parametric equations are:

\[
x = t
\]
\[
y = 0
\]

---

### **56. Parametric Equations in \( \mathbb{R}^n \):**

#### **(a) Line passing through the points \((-1, 0, 3, -12)\) and \((6, -15, 11, -8)\) in \( \mathbb{R}^4 \):**

The direction vector \( \mathbf{v} \) is:

\[
\mathbf{v} = (6 - (-1), -15 - 0, 11 - 3, -8 - (-12)) = (7, -15, 8, 4)
\]

Now, the parametric equations through \((-1, 0, 3, -12)\) are:

\[
x = -1 + 7t
\]
\[
y = 0 - 15t
\]
\[
z = 3 + 8t
\]
\[
w = -12 + 4t
\]

#### **(b) Line whose slope-intercept form is \( y = -\frac{1}{3}x - \frac{5}{3} \) in \( \mathbb{R}^2 \):**

Here, the slope \( m = -\frac{1}{3} \) and the y-intercept is \( -\frac{5}{3} \). The parametric equations are:

\[
x = t
\]
\[
y = -\frac{1}{3}t - \frac{5}{3}
\]

#### **(c) Line along the x-axis in \( \mathbb{R}^3 \):**

For a line along the x-axis in \( \mathbb{R}^3 \), the \( y \)- and \( z \)-coordinates are always 0, and \( x \) varies freely. The parametric equations are:

\[
x = t
\]
\[
y = 0
\]
\[
z = 0
\]

---

### Would you like any details or have questions on this? Here are 5 related questions:

1. How do we determine the direction vector between two points?
2. What is the difference between parametric and symmetric equations of a line?
3. Can a line in \( \mathbb{R}^n \) be represented by more than one set of parametric equations?
4. How does the parametric equation of a line differ when the slope is given?
5. How do parametric equations generalize for higher dimensions like \( \mathbb{R}^4 \)?

**Tip**: In parametric equations, the parameter \( t \) typically represents how far along the line you move, making it a powerful tool for visualizing and working with lines in multiple dimensions.

55. Write parametric equations of each given line: a) the line in R^3 passing through (11, −6, 7) parallel to vector 3i + j − 12k b) the line in R^3 passing through the points (3, 1, 3) and (2, −4, −2) c) the x-axis (in R^2) 56. Write parametric equations of each given line: a) the line in R^4 passing through the points (−1, 0, 3, −12) and (6, −15, 11, −8) b) the line in R^2 whose slope-intercept equation is y = −1/3x − 5/3 c) the x-axis (in R^3)

Learn how to write parametric equations for lines in 3D and 4D space. Explore problems involving points, direction vectors, and slope-intercept forms. Suitable for advanced high school and undergraduate students studying vector geometry and parametric equations.

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