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:

$x = t$ $y = 0$

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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:

$x = t$ $y = 0$ $z = 0$

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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.