To solve question 11, we need to find the parametric and symmetric equations of the line through the point $(1, -1, 1)$ that is parallel to the line given by the equations:

$x + 2 = \frac{1}{2}y = z - 3$

Step 1: Interpret the Given Line

The symmetric equation $x + 2 = \frac{1}{2}y = z - 3$ represents a line. We can separate it to extract the direction vector of the line.

Rewrite the symmetric form into individual parametric forms:

1. $x = t - 2$
2. $\frac{1}{2}y = t \Rightarrow y = 2t$
3. $z = t + 3$

From these parametric equations, we can deduce that the direction vector $\vec{d}$ of this line is: $\vec{d} = \langle 1, 2, 1 \rangle$

Step 2: Define the New Line

The new line we need to find goes through the point $(1, -1, 1)$ and is parallel to the direction vector $\langle 1, 2, 1 \rangle$.

Parametric Equations

Using the point $(1, -1, 1)$ and direction vector $\langle 1, 2, 1 \rangle$, the parametric equations for the line are: $x = 1 + t$ $y = -1 + 2t$ $z = 1 + t$

Symmetric Equations

To convert these parametric equations into symmetric form, we solve each equation for $t$:

1. From $x = 1 + t$, we get $t = x - 1$.
2. From $y = -1 + 2t$, we get $t = \frac{y + 1}{2}$.
3. From $z = 1 + t$, we get $t = z - 1$.

Thus, the symmetric form of the line is: $\frac{x - 1}{1} = \frac{y + 1}{2} = \frac{z - 1}{1}$

Final Answer

The parametric equations are: $x = 1 + t, \quad y = -1 + 2t, \quad z = 1 + t$

The symmetric equation is: $\frac{x - 1}{1} = \frac{y + 1}{2} = \frac{z - 1}{1}$

Would you like more details on any of the steps?

Here are five related questions to explore further:

1. How do you convert a symmetric equation of a line back into parametric form?
2. What if a line passes through a different point but remains parallel to the same direction vector?
3. How can you determine if two lines in space are parallel?
4. How would you find the point of intersection between two non-parallel lines?
5. What is the significance of the direction vector in defining the orientation of a line?

Tip: To find the direction vector from a symmetric equation, compare each term’s coefficient to the parameter $t$ when rewriting as parametric equations.